theoretical study of thermal analysis kinetics

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Theses and Dissertations--Mechanical Engineering Mechanical Engineering

2014

THEORETICAL STUDY OF THERMAL ANALYSIS KINETICS THEORETICAL STUDY OF THERMAL ANALYSIS KINETICS

Yunqing Han University of Kentucky, [email protected]

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REVIEW, APPROVAL AND ACCEPTANCE REVIEW, APPROVAL AND ACCEPTANCE

The document mentioned above has been reviewed and accepted by the student’s advisor, on

behalf of the advisory committee, and by the Director of Graduate Studies (DGS), on behalf of

the program; we verify that this is the final, approved version of the student’s thesis including all

changes required by the advisory committee. The undersigned agree to abide by the statements

above.

Yunqing Han, Student

Dr. Kozo Saito, Major Professor

Dr. James M McDonald, Director of Graduate Studies

THEORETICAL STUDY OF THERMAL ANALYSIS KINETICS

Dissertation

A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the

College of Engineering at the University of Kentucky

By Yunqing Han

Lexington, Kentucky Director: Dr. Saito, Professor of Mechanical Engineering

Lexington, Kentucky 2014

Copyright © Yunqing Han 2014

In the past decades, a great variety of model fitting and model free (isoconversional) methods have been developed for extracting kinetic parameters for solid state reactions from thermally stimulated experimental data (TGA, DSC, DTA etc.). However, these methods have met with significant controversies about their methodologies. Firstly, model-fitting methods have been strongly criticized because almost any reaction mechanism can be used to fit the experimental data satisfactorily with drastic variations of the kinetic parameters, and no good criterion exists to tell which mechanism is the best choice. Secondly, previous model free methods originated from the isoconversional principle, which is often called the basic assumption; previous studies comparing the accuracy of model free methods have not paid attention to the influence of the principle on model free methods and, therefore, their conclusions are problematic. This work gives, firstly, a critical study of previous methods for evaluating kinetic parameters of solid state reactions and a critical analysis of the isoconversional principle of model free methods. Then an analysis is given of the invariant kinetic parameters method and recommends an incremental version of it. Based on the incremental method and model free method, a comprehensive method is proposed that predicts the degree of the dependences of activation energy on heating programs, and obtains reliable kinetic parameters. In addition, this work also compares the accuracy of previous methods and gives recommendations to apply them to kinetic studies.

KEYWORDS: Thermal Analysis Kinetics, Model Fitting method, Model Free Method, Comprehensive Method, Activation Energy

Date

Student’s Signature

ABSTRACT OF DISSERTATION

THEORETICAL STUDY OF THERMAL ANALYSIS KINETICS

Yunqing Han

05-08-2014

Approval Page

STUDY ON METHODOLOGY OF THERMAL ANALYSIS

KINETICS

By

Yunqing Han

05-08-2014

Director of Dissertation

Dr. Kozo Saito

Director of Graduate Studies

Dr. James McDonough

To my mother, Fang Li

Your support, encouragement, and endless dedication have sustained me throughout my life.

iii

ACKNOWLEDGMENTS

I would like to especially express my sincere gratitude to my advisor Dr. Kozo Saito for his

guidance in both academic progress and attitude toward life. I am also thankful to my co-advisor

Dr. Tianxiang Li for his support and helpful discussions. I am also grateful to Dr. John Stencel for

his patience and instructive comments while reviewing my work. I am thankful for the

encouragement provided by both Dr. Nelson Akafuah and Dr. Ahmad Salaimeh during the

preparation of this dissertation. And I am also greatly thankful to the external examiner Dr. Peter

Hislop, and the entire dissertation committee: Dr. Dusan Sekulic, Dr. Tate Tsang, Dr. David

Herrin, Dr. José Graña-Otero for their invaluable discussions.

I also would like to thank to Ms. Brittany Adam, Mr. Justin English, Mr. Matthew Lane, Mr.

Nicholas Cprek, Mr. Jeremy Fugate, Ms. Madison Donoho, Ms. Kendall Kruszewski, Mr. Greg

Brotzge, Mr. Brent Hale, Mrs. Rebecca Hale, Mr. Philip Duff, Ms. Emily Woods and Ms. Corrie

Mckee, for their company, friendship, and especially the time they dedicated in correcting my

pronunciations. Without their help, the tremendous improvement in my English efficiency would

not have been possible. They also made my stay at the University of Kentucky enjoyable and

made the three years fly by; I will fondly look back on those memories.

I also would like to thank my friends especially, Shaoqian Wang, Yulong Yao, Qiao Liang, Hao

Zhang for their friendship and encouragement. I would also like to thank my roommate Jian

Zhang, who has greatly broadened my vision to living philosophy and motivated my ambitious to

personal success.

Specially, I also would like to thank the warmhearted, sincere, and conscientious lady, Ms. Claire

A. Adams. There are questions, such as the meaning of death and life, and what on earth is love,

that always confused me and they never seem to have standard answers. As a person who wants

but does not have a religion at this time, it seems every person could be a teacher, and I do not

know where my final spiritual destination is. However, her existence has shown me how peaceful

iv

and joyful it is to be a person of devout faith and it is worthy to calm down and listen to my heart,

not just to make a life.

Lastly, my appreciation goes to Robert Frost, who wrote my favorite English poem:

The Road not Taken

Two roads diverged in a yellow wood

and sorry I could not travel both

And be one traveller, long I stood

and looked down one as far as I could

to where it bent in the undergrowth;

Then took the other, as just as fair,

and having perhaps the better claim

because it was grassy and wanted wear;

though as for that, the passing there

had worn them really about the same,

And both that morning equally lay

in leaves no feet had trodden black.

Oh, I kept the first for another day!

Yet knowing how way leads on to way,

I doubted if I should ever come back.

I shall be telling this with a sigh

Somewhere ages and ages hence:

Two roads diverged in a wood, and I --

I took the one less travelled by,

and that has made all the difference

v

TABLE OF CONTENTS

ACKNOWLEDGMENTS .............................................................................................................. iii

LIST OF FIGURES ....................................................................................................................... vii

LIST OF TABLES .......................................................................................................................... ix

CHAPTER 1: Introduction ........................................................................................................ 1

1.1 Kinetics of Thermally Stimulated Solid Reactions .......................................................... 1

1.2 Kinetic Triplets and Equations ......................................................................................... 3

1.3 Outline of Dissertation ..................................................................................................... 7

CHAPTER 2: PREVIOUS METHODS AND COMPARISONS ........................................... 9

2.1 General Equation and Temperature Integral .................................................................... 9

2.2 Model Fitting Methods .................................................................................................. 10

2.3 Invariant Kinetic Parameters Method ............................................................................ 15

2.4 Model Free Method ........................................................................................................ 16

2.4.1 Isoconversional Principle ........................................................................................... 16

2.4.2 Differential Isoconversional Methods ........................................................................ 17

2.4.3 Regular Integral Methods........................................................................................... 18

2.4.3.1 Ozawa-Flynn-Wall Method ....................................................................................... 18

2.4.3.2 Vyazovkin Method ..................................................................................................... 19

2.4.3.3 Li-Tang Method ......................................................................................................... 20

2.4.3.4 Kissinger-Akahira-Sunose Method ............................................................................ 21

2.4.4 Advanced Integral Methods ....................................................................................... 22

2.4.4.1 Advanced Vyazovkin Method ................................................................................... 22

2.4.4.2 Advanced Li-Tang Method ........................................................................................ 23

2.4.5 Modulated Thermogravimetry Methods .................................................................... 24

CHAPTER 3: MODIFICATION OF REGULAR LI-TANG METHOD AND ORTEGA METHOD ..................................................................................................................... 25

3.1 Advanced Li-Tang Method ............................................................................................ 26

3.1.1 Methodology of Advanced Li-Tang Method ............................................................. 26

3.1.2 Numerical Applications ............................................................................................. 29

3.1.3 Experimental Example ............................................................................................... 32

3.2 Modified Ortega Method ............................................................................................... 33

3.2.1 Methodology of Modified Ortega Method ................................................................. 33

3.2.2 Results and Discussion .............................................................................................. 37

vi

3.2.2.1 Simulation without Noise........................................................................................... 37

3.2.2.2 Simulation with Noise ................................................................................................ 39

3.2.2.3 Experimental Application .......................................................................................... 40

3.3 Conclusions .................................................................................................................... 41

CHAPTER 4: COMPREHENSIVE METHOD BASED ON MODEL-FREE AND IKP METHODS TO EVALUATE KINETIC PARAMETERS ...................................... 43

4.1 Critical Analysis of Model Free Methods ...................................................................... 44

4.2 Analysis of IKP Method ................................................................................................ 46

4.3 Proposition of a Comprehensive Method ....................................................................... 47

4.4 Simulation Validations ................................................................................................... 54

4.5 Experimental Validation ................................................................................................ 59

4.6 Conclusion ..................................................................................................................... 61

CHAPTER 5: ACCURACY OF ISOCONVERSIONAL METHODS WITH A CONSIDERATION OF BASIC ASSUMPTION ...................................................... 63

5.1 Previous Comments about Isoconversional Methods .................................................... 63

5.2 Error Sources of Isoconversional Methods .................................................................... 64

5.3 Simulations and Analysis ............................................................................................... 67

5.3.1 Parallel Independent Reaction ................................................................................... 67

5.3.2 Parallel Competitive Reaction ................................................................................... 72

5.4 Conclusion ..................................................................................................................... 75

CHAPTER 6: Conclusion ........................................................................................................ 77

References ........... ......................................................................................................................... 79

Vita…….... ......... .......................................................................................................................... 91

vii

LIST OF FIGURES

Figure 1.1 Schematic of TG and DSC plots……………………………………………….. ….2

Figure 3.1 Eα dependencies evaluated for the simulated process by Li-Tang method with

different lower limit of the integral, αl, as well as by OFW, AIC, FR and the new

method .................................................................................................................... 30

Figure 3.2 Eα dependencies evaluated for the SrCO3 decomposition by the new method, FR

method and AIC method ......................................................................................... 31

Figure 3.3 Eα dependencies obtained for the SrCO3 decomposition by the new method and Li-

Tang method with different lower integral limit,αl , aand by OFW method .......... 32

Figure 3.4 Dependence of Eα on the value α evaluated by FR, AIC, OFW, the original Ortega

method, and the modified Ortega method with ∆α=0.04 ........................................ 36

Figure 3.5 Dependence of Eα on the value α for the simulated single step reaction FR, AIC,

and the original Ortega method with ∆α =0.01, and the modified Ortega method

with ∆α =0.04 .......................................................................................................... 38

Figure 3.6 Dependence of Eα on the value α obtained for the experiments of SrCO3

decomposition by AIC, the modified Ortega method, and the original Ortega

method, all with ∆α =0.04 ....................................................................................... 39


decomposition by AIC, FR and Ortega method all with ∆α=0.04 .......................... 40

Figure 4.1 The dependences of E and lnA on α of a single step reaction determined by IIKP

method .................................................................................................................... 46

Figure 4.2 Schematic of IIKP principle for complex reactions ................................................ 48

Figure 4.3 Flow chart of comprehensive method ..................................................................... 50

viii

Figure 4.4 Values of activation energy determined by FR and IIKP methods for simulation

test S1 ...................................................................................................................... 41


test S2 ...................................................................................................................... 52


test S3 ...................................................................................................................... 52


test S4 ...................................................................................................................... 53


test S5 ...................................................................................................................... 53


test S6 ...................................................................................................................... 54


test S7 ...................................................................................................................... 56

Figure 4.11 Values of activation energy obtained by different methods for experiment data ... 60

Figure 4.12 Values of selected activation energy determined by comprehensive method ......... 60

Figure 5.1 Simulation results obtained from test T1, (beta1-3 are true values for different β i: it

also applies all Figures) .......................................................................................... 66

Figure 5.2 Simulation results obtained from test T2 ................................................................ 67








ix

LIST OF TABLES

Table 1.1 Expressions of functions of the most common reaction mechanisms ......................... 5

Table 4.1 The values of E_inv and A_inv obtained by IIKP method ....................................... 57

1

CHAPTER 1: INTRODUCTION

1.1 Kinetics of Thermally Stimulated Solid Reactions

Thermal activation is, probably, the most common means to stimulate solid state reactions,

although the applications of photo activation, magnetic field, pressure, and electrochemical

potentials are also possible. By activating by external heating or cooling stimuli, the structure,

phase state, and chemical properties of solids are changeable; thermal analysis techniques

measure the physical and chemical changes of solids as a function of temperature in controlled

thermal conditions. Thermal analysis techniques have been employed since the early 20th century

and are increasingly important as an analytical tool in the fields of chemistry, physics, materials,

geology, metallurgy, medicine, and combustion.

The development of thermal analytical instruments and thermal analysis methods have provided a

useful tool to obtain the kinetic parameters of solid state reactions with a small amount of solid

sample. The most common and widely used thermal analysis techniques are Thermogravimetric

Analysis (TGA) and Differential Scanning Calorimetry (DSC). TGA is a method commonly used

to measure selected characteristics of materials’ mass changes due to decomposition, oxidation,

or loss of volatiles (e.g., moisture and combustibles) and to record information digitally as a

function of increasing temperature and/or of time (Fig. 1.1). Most typical TGA applications are

studies of reaction kinetics and degradation mechanisms, materials characterization by analysis of

characteristic decomposition patterns, and determination of organic or inorganic contents in

samples. DSC is a thermoanalytical technique that measures the difference in the amount of heat

needed to increase the temperature of a sample and a reference as a function of temperature. The

fundamental mechanism underlying this technique is that, more or less, the reactions will be

exothermic or endothermic, and heat will need to flow to or from the sample and reference to

keep them at the same temperature when the sample experiences a physical transformation. For

instance, a phase transition from solid to liquid absorbs heat; when a solid sample melts, it will

2

take more heat to increase its temperature at the same rate as the reference, and DSC is able to

measure the amount of heat absorbed or released during the reaction (Fig. 1.1). DSC is also

applied to observe more subtle physical changes (e.g., glass transitions and polymer curing). A

similar technique is differential thermal analysis (DTA) in which heat flows to the reference and

the sample and is kept the same rather than the temperature. Hence, DSC and DTA provide

similar information. Solid state kinetic data obtained by TGA and DSC are of an increasing

practical interest because a growing number of technologically important processes like thermal

energetic materials and crystalline solids, thermal oxidation and pyrolysis of fuels and polymers,

crystallization of glasses and polymers, and the solidification of metallic alloys are fruitfully

studied using these techniques [1].

Figure 1.1 Schematic of TG and DSC plots

Thermal Analysis Kinetics (TAK) seeks to quantitatively analyze the relationships between

temperature and physical properties (e.g., the mass change as a function of time) measured by the

thermal analysis techniques. The development of TAK is based on chemical thermodynamics,

3

chemical kinetics and thermal analysis techniques. By analyzing data obtained by thermal

analysis techniques, TAK is able to provide kinetic parameters, estimate the thermal stability and

life span of materials, and the best operation conditions of polymers, quantitatively describe the

reaction rate and reaction mechanisms, and provide supporting information for estimating

properties of energetic materials and combustibles [2].

Interests in TAK were awakened in the early 20th century, and tremendous developments

occurred during the recent decades. TAK has been developed for no less than one hundred

analytical methods and applied in various fields. It is capable of quantitatively characterizing

reactions and phase change processes, determining the most probable reaction mechanisms, and

extracting activation energies and pre-exponential factors of solid state reactions.

1.2 Kinetic Triplets and Equations

Non-isothermal, heterogeneous thermal analysis kinetics originated from the theory of isothermal

and homogenous gas or liquid phase kinetics, the basis of which had been established by the end

of 19th century. Its description equation is

( ) ( )cfTkdtdc

= (1.1)

where c is the concentration, t is the time, T is temperature, k(T) is the rate constant that

dependent on temperature, and ( )cf is the reaction mechanism. In isothermal and homogenous

reactions it is often represented by ( ) ( )nccf −= 1 .

Early solid state kinetic studies were carried out under isothermal conditions [3-5], and used the

following kinetic equation

( ) ( )αα fTkdtd

= (1.2)

4

whereα is the extent of reaction expressed by

fmmmm

−−

=0

0α (1.3)

Expression 1.3 is analogous to molar concentrations of gas reactants and/or products, of which m

is the mass of the reactant, the subscripts 0 and f designate the initial and final states, respectively,

and ( )αf is the reaction model related to the solid reaction mechanism. Unlike in gas, molecular

motion is highly restricted in solids and reactions are dependent on local structure and activity;

some of the models are derived strictly according to their mechanistic basis such as nucleation,

geometrical contraction, diffusion, and reaction order [6, 7]. -Most common reaction models are

listed in Table 1.1.

The temperature dependence of the rate of solid state reactions is typically parameterized through

the Arrhenius equation [8]

( ) ( )RTEATk /exp −= (1.4)

where A is the pre-exponential factor, E is the activation energy, and R is the universal gas

constant.

The use of the Arrhenius equation to parameterize temperature dependence has generated

problems of interpreting experimentally determined values of E and A, and have been criticized

from a physical standpoint [9, 10]. Reference [10] has stressed that the Arrhenius equation is only

meaningfully applicable to reactions that take place under homogeneous conditions. However, as

pointed out in Ref. [11], thermal decomposition has been demonstrated successfully [12, 13] in

the framework of an activated theory from an Arrhenius-like equation for the temperature

dependence of the process. Moreover, the Arrhenius equation is useful for describing ( )Tk of

many thermally activated, heterogeneous solid state reactions such as diffusion [14], nucleation

5

and nuclei growth [15], presumably because the system has to overcome an energy barrier and the

energy distribution along the relevant coordinate is controlled by Boltzmann statistics. In

addition, Galwey and Brown [16] have demonstrated that the statistics of Fermi-Dirac and Bose-

Einstein give rise to Arrhenius-like equations, even in cases where the density of available state is

sparse. Therefore, Ref. [11] concluded that the use of the Arrhenius equation is justifiable in terms

of a rational parameterization, and its use and physical interpretation are on a sound theoretical

basis.

Table 1.1 Expressions of functions of the most common reaction mechanisms

Number Model Differential form 𝑓(𝛼) Integral form 𝐺(𝛼) Nucleation models

1 Power law 𝑃1 1 𝛼 2 Power law 𝑃3/2 (2/3)𝛼−1/2 𝛼2/3 3* Power law 𝑃2 2𝛼1/2 𝛼1/2 4* Power law 𝑃3 3𝛼2/3 𝛼1/3 5* Power law 𝑃4 4𝛼3/4 𝛼1/4

Sigmoidal rate equations 6 Avarami-Erofe’ev𝐴3/2 (3/2)(1 − 𝛼)[−𝑙𝑛(1 − 𝛼)]1/3 [−𝑙𝑛(1 − 𝛼)]2/3 7* Avarami-Erofe’ev𝐴2 2(1 − 𝛼)[−𝑙𝑛(1 − 𝛼)]1/2 [−𝑙𝑛(1 − 𝛼)]1/2 8* Avarami-Erofe’ev𝐴3 3(1 − 𝛼)[−𝑙𝑛(1 − 𝛼)]2/3 [−𝑙𝑛(1 − 𝛼)]1/3 9* Avarami-Erofe’ev𝐴4 4(1 − 𝛼)[−𝑙𝑛(1 − 𝛼)]3/4 [−𝑙𝑛(1 − 𝛼)]1/4 10 Prout-Tomkins 𝐴𝑢 𝛼(1 − 𝛼) 𝑙𝑛⌊𝛼/(1 − 𝛼)⌋

Geometrical contraction models

11* Contracting area R2 2(1 − 𝛼)1/2 1 − (1 − 𝛼)1/2 12* Contracting volume R3 3(1 − 𝛼)2/3 1 − (1 − 𝛼)1/3

Diffusion models 13* 1D Diffusion 𝐷1 1/2𝛼 𝛼2 14* 2D Diffusion 𝐷2 [−𝑙𝑛(1 − 𝛼)]−1 (1 − 𝛼)𝑙𝑛(1 − 𝛼) + 𝛼 15* 3D Diffusion-Jander𝐷3 (3/2)(1 − 𝛼)2/3 1 − (1 − 𝛼)1/3 1 − (1 − 𝛼)1/32

16* Ginstling-Brounshtein𝐷4 (3/2)/(1 − 𝛼)−1/3 − 1 1 − (2𝛼/3)− (1 − 𝛼)2/3

17 Zhuravlev, lesokin, Tempelman𝐷5

(3/2)(1 − 𝛼)4/3/(1 − 𝛼)−1/3

− 1 (1 − 𝛼)−1/3 − 12

18 Anti-Jander𝐷6 (3/2)(1 + 𝛼)2/3/(1 + 𝛼)1/3 − 1 (1 + 𝛼)1/3 − 12 Reaction-order models

19 One-third order 𝐹1/3 (3/2)(1 − 𝛼)1/3 1 − (1 − 𝛼)2/3 20 Three-quarters order 𝐹3/4 4(1 − 𝛼)4/3 1 − (1 − 𝛼)1/4 21 One and a half order 𝐹3/2 2(1 − 𝛼)4/2 (1 − 𝛼)−1/2 − 1 22* First-order 𝐹1 1 − 𝛼 −𝑙𝑛(1 − 𝛼) 23* Second-order 𝐹2 (1 − 𝛼)2 (1 − 𝛼)−1 − 1 24* Third-order 𝐹3 (1 − 𝛼)3 [(1 − 𝛼)−2 − 1]/2

6

As pointed out in Ref. [11], Vallet studied the first kinetic evaluations of non-isothermal data that

were carried out at a constant heating rate, dtdT /=β . To extract values of the kinetic

parameters, Vallet suggested replacing the temporal differential in Eq. (1.2) by

β/dTdt = (1.5)

Note the transformation of Eq. (1.5) implicitly contains an assumption that the change in

experimental conditions from isothermal to non-isothermal does not affect reaction kinetics; this

assumption may have serious implications for multi-step reaction kinetics [11].

Based on the aforementioned theories, the equation of heterogeneous solid state reaction rate

under isothermal condition can be described as

( )αα fRTEA

dtd

−= exp (1.6)

which under non-isothermal conditions with a constant heating rate leads to

( )αβ

α fRTEA

dTd

−= exp (1.7)

The parameters of E, A, and ( )αf are often called the kinetic triplet, which are to be determined

during the kinetic analyses of solid state reactions.

Currently, the core of TAK is to study the kinetics of non-isothermal solid state reactions

(including physical effects). The reason for using non-isothermal conditions is because of the

difficulty to attain strict isothermal conditions, especially during the initial stage of a reaction

process; using isothermal conditions is also more time consuming. Moreover, theoretically, a

thermal experimental curve obtained under non-isothermal conditions could carry information

equivalent to that in multiple data curves obtained from isothermal conditions.

7

Almost all currently existing methods start from Eq. (1.6) and Eq. (1.7), but quite different results

for the kinetic values are often given even by different researchers for a simple reaction. A typical

example is the activation energy for dehydrating calcium oxalate monohydrate (C2H2CaO5), a

representative example of a single-step reaction: literature values for activation energy range

from less than 50 kJ/mol to more than 200 kJ/mol [17]. This wide variation has been shown to be

influenced by experimental conditions [18]. However, none of the currently available thermal

analysis techniques is capable of providing experimental data without important influences from

the experimental conditions, even with strict control of the heating program, sample size, and

initial mass. Especially, disparate heating programs that are required for some methods may affect

solid-state kinetics by influencing physical processes of the reaction, like diffusion, adsorption,

and desorption of gaseous products or reactants from the solid surface. Therefore, effects of

experimental conditions, especially of the heating program, on the apparent model function

should be examined extensively. Recall that current solid-state kinetic theory came from classical

kinetic concepts of homogeneous reactions, and the usage and interpretation of the Arrhenius

equation in solid state kinetics were supported by both empirical tests and theoretical

examinations [19-21]. In homogeneous reactions, the thermodynamic meaning of activation

energy is the heat absorbed in the process of transforming inactive molecules into active ones.

In addition, more than one hundred methods have been developed with a great difference in both

methodology and applicability, and the accuracy and reliability of them needs to be carefully

examined. More importantly, a comprehensive method needs to be developed that surmounts the

influences of experimental procedures and enables simplification of them, while offering more

accurate and repeatable kinetic parameters.

1.3 Outline of Dissertation

This study contains six (6) chapters, the structure of which is organized as follows.

8

Chapter 1 introduces the study of solid state thermal analysis kinetics, the development of a

general kinetic equation, and the task of thermal analysis kinetics.

Chapter 2 presents, methodically, a review of previous methods for evaluating kinetic parameters

of solid state reactions, including the model fitting method, invariant kinetic parameter method,

and model free methods.

Chapter 3 uses two specific examples to give a detailed analysis of previous methods, which is

useful for better understanding about how previous studies compare the accuracy of model free

methods, and their achievements and problems.

Chapter 4 is the main focus of this study. Firstly, it shows theoretically that the activation energy

for complex reactions is both functions of reaction degree and heating programs. Model free

methods that try to extract dependences of activation energy on conversion degree without

considering the dependences of heating programs are problematic. Then, an analysis of the

invariant kinetic parameters method is presented and discussed, and an incremental version of it

is described. Based on the incremental, invariant kinetic parameters method and model free

method, a comprehensive method is proposed that predicts the degree of the dependencies of

activation energy on heating programs, selects reliable values of activation energy, and extracts

values of the variable pre-exponential factor.

Chapter 5 gives an additional analysis of the accuracy of previous model free methods by

considering the influence of the isoconversional principle.

Finally, Chapter 6 makes a conclusion of this study.

9

CHAPTER 2: PREVIOUS METHODS AND COMPARISONS

In the past decades, a variety of methods have been developed for extracting single pair of or

variable kinetic parameters for solid state reactions from thermal stimulated experimental data

(TGA, DSC, DTA etc.) that could be unified and visualized as the conversion curves of α and/or

/d dTα as a function of temperature or time. These methods have been categorized generally

into model fitting methods and model free methods.

2.1 General Equation and Temperature Integral

Nearly all the thermal analysis methods start from the general differential kinetic Eqs. (1.6) and

(1.7) or the integral forms of them,

( ) ( ) ( )0 0

1 expt Eg d A dt

f RT tα

α αα

= = −

∫ ∫ (2.1)

For a linear heating program, dtdT /=β , the above equation leads to

( ) ( )20exp

xT

x

A E AE e AEg dT dx p xRT R x R

αβ β β

−∞ = − = = ∫ ∫ (2.2)

where “x” denotes E/RT, and p(x) is the temperature integral. Note that the derivation of the

above equation involves an assumption that E must be a constant with respect toα .

For a specific value x, the temperature integral, p(x), has no analytical solution but has been

approximated by hundreds of possibilities [22-27]. Doyle [22-24] suggested a linear

approximation of the logarithm of p(x),

( )log = 0.4567 2.315p x x− − (2.3)

which is equivalent to

10

( ) ( )= exp 1.0518 5.330p x x− − (2.4)

The approximation given by Coats and Redfern [25] is,

( ) ( )2

exp 2= 1x

p xx x− −

(2.5)

A more popular, fourth order approximation is given by Senum and Yang [26],

( ) ( ) 4 3 2

2 4 3 2

exp 18 86 96=20 120 240 120

x x x x xp xx x x x x− + + +

+ + + + (2.6)

Analyses of accuracies of the approximations have been discussed [28, 29], with the conclusion

that the fourth order Senum and Yang approximation is the most accurate. Other approximations

result in greater errors although they may be more convenient to apply than the Senum and Yang

approximation. It should be kept in mind that these analyses are based on the assumption that the

value of /x E RT= is a constant.

2.2 Model Fitting Methods

Model-fitting methods are the major methods used during analyses of thermal analysis kinetics in

which TG and DSC experimentation determine mass or heat change as a function of temperature

or time. A model-fitting method is a one that fits different reaction models ( )f α or ( )g α into

the general kinetic equation, Eq. (1.7) or Eq. (2.2), and values of the activation energy and pre-

exponential factor are calculated by regression analyses [1]. Then, different groups of kinetic

triplet values are used to fit one of the above equations, and the curves generated by the equation

with the best to the actual experimental curves would be considered the proper one to be selected.

For brevity, all equations in this study are derived with a linear heating rate because it is the most

commonly used approach, although an arbitrary heating program can be derived by replacing

( )1/ /d dTβ α with /d dtα . In the following, the Gorbatchev [30] and Coats-Redfern methods

11

[25] are presented and discussed to give a better understanding of the detailed procedure

associated with model-fitting methods.

From the temperature integral ( )p x , one gets:

( ) 2

xu ep x dxx

−

∞

−= ∫

= 22

x xx xe e dxx

−− −

∞ ∞− ∫

= 32 2x x xe x de

x

−− −

∞− ∫

= 42 3

2 ( 6)x xx xe e e x dx

x x

−− − −

∞− + −∫

= 42 3

2 6x xx xe e x de

x x

−− − −

∞− + ∫

= 2 3 4 4

2 6 6x xx x x xe e e e dx x x x

−− − −

∞ ∞− + − ∫

=…=

2 2 3

2! 3! 41 ...xe

x x x x

− − + − +

！ (2.7)

Then Eq. (2.2) leads to

2 2 30

2! 3! 4exp 1 ...xT E E edT

RT R x x x x

− − = − + − + ∫

！

(2.8)

Using the first two terms leads to the following approximation of the temperature integral

12

2

0

2exp 1 expT E ET RT EdT

RT R E RT − = − − ∫ (2.9)

Equation 2.9 is the Coats-Redfern approximation.

Multiplying the term of ( )1 2 /RT E+ to both sides gives,

22

0

21exp exp21

T

ET RTR EE EdT RTRT RT

E

− − = − +

∫ (2.10)

In most cases T is in a moderate range and E is bigger than 60kJ/mol so that the term

2 / 1RT E << and equation 2.10 simplifies to

2

2

0

expexp exp2 21

T

ET EE ET ER RTdT RTRT E RT RT

E

− − = = − + +∫ (2.11)

This equation is the Gorbatchev approximation.

Combining Eq. (2.2) and Eq. (2.11) leads to

( )

( )2ln ln2

g AR ET E RT RTα

β

= − + (2.12)

If we approximate the first term on the right hand side as a constant, for a proper ( )g α a plot of

( ) 2ln /g Tα and 1 / T will be a straight line or linear function from which can be obtained the

values of E and A from the slope and (0,0) intersect, respectively. If the first term cannot be

approximated as a constant, Eq. (2.12) can be transformed to

13

( )( )

2

2ln ln

g E RT AR ET RT

αβ

+ = −

(2.13)

Then use of iteration and the least square method enables the calculation of E、A and a logically

reasonable ( )g α . This approach is called the Gorbatchev method [30].

If Eq. (2.2) and Eq. (2.9) are combined, and set ( ) ( )1 nf α α= − while using a first order

approximation of ( )P x , the following equation is obtained

( )

2

0

21 exp1 n

d A RT RT EE E RT

α αβα

= − − −∫ (2.14)

Taking the logarithm of both sides (if 1n ≠ ) gives,

( )( )

1

2

1 1 2ln ln 11

nAR RT E

T n E E RTα

β

− − − = − − − (2.15)

If 1n = , then

( )

2

ln 1 2ln ln 1AR RT ET E E RT

αβ

− − = − − (2.16)

The above two equations are the Coats-Redfern method. Since in most cases / 1E RT >> ,

(1 2 / ) 1RT E− ≈ and the first term on the right hand side of the above two equations is

approximately a constant. Plotting ( )( ) ( )( )1 2ln 1 1 / 1n T nα − − − −

with respect to 1 / T when

1n ≠ or plotting ( )( ) 2ln ln 1 / Tα − − with respect to 1 / T when 1n = , a straight line is

obtained if the value of n is chosen properly and the value of the activation energy is obtained

14

from the slope. In cases where the above two equation fails to satisfy (1 2 / ) 1RT E− ≈ , the

following evaluation functions may be applied:

[ ]2

1.3.15 .3.15

L

iLHS termof Eq RHS terms of Eq

=

Ω = −∑

[ ]2

1.3.16 .3.16

L

iLHS termof Eq RHS terms of Eq

=

Ω = −∑

where LHS and RHS designate left hand side and right hand side, respectively.

The use of the above two evaluation functions to calculate a minimum value then gives the values

of E, A and n. If 1n ≠ , another procedure is to transform Eq. (2.11) and Eq. (2.12) to,

( )

( )

1

2

1 1ln ln

21 1

nAR E

RT E RTT nE

αβ

− − − = − − −

(2.17)

If 1n = , the following equation is obtained,

( )2

ln 1ln ln

21

AR ERT E RTTE

αβ

− = − −

(2.18)

The values of E, A and n can be obtained by applying an iteration or least squares method, the

approach of which is called the Coats-Redfern method [25, 31]. If the value of n is close to 0, Eq.

(2.18) leads to

2

2ln ln 1AR RT ET E E RTα

β = − −

(2.19)

15

Then, a plot of ( )2ln / Tα against 1 / T gives the value of E from the slope and the value of A

from the (0, 0) intercept.

Combining Eq. (2.2) and the first order approximation of the temperature integral,

( ) 2/xp x e x−= , and doing some transformations give another form of Coats-Redfern integral,

( )

2ln lng AR ET E RTα

β

= −

(2.20)

A plot of ( )( )2ln /g Tα as a function of 1 / T gives the values of E and A from the slope and

intercept, respectively. [2]

Though a great number of model fitting methods have been developed by researchers, the

principle of these methods is the same as in the discussion heretofore: fit different reaction

models into the kinetic equation and calculate the activation energy and pre-exponential factor,

and then select the model which gives the best fit. All of these model fitting methods give a single

pair of values for the activation energy and pre-exponential factor. In general, model-fitting

methods have been strongly criticized [2, 32-38] because almost any ( )f α can be used to fit the

experimental data satisfactorily even though drastic variations in the kinetic parameters occur; no

good criterion exists to distinguish which result best reflects or describes actual physiochemical

processes occurring during the reactions..

2.3 Invariant Kinetic Parameters Method

Model-fitting methods involve the use of different reaction models to fit one single conversion

curve or multiple curves, and then attempt to determine the kinetic parameters E and A by

regression analyses. When a model-fitting method is applied to a single-heating rate test, widely

varying values of the activation parameters are obtained when using different model functions

16

that can be correlated to the so-called “compensation effect” relation [39-45]:

lnA a bE= + (2.21)

where a and b are constants. It has been theoretically and experimentally postulated that

1/ maxb RT= and in some literature, 2/ maxa E RTβ= , where the index max means the

maximum reaction rate [45-50].

The invariant kinetic parameter (IKP) method, suggested by Lesnikovich and Levchik [40, 46,

51], employs the compensation effect. Since the linear regression lines, represented by Eq. (2.21),

for several sets of different heating rates, jβ , tend to intersect at a point or a narrow common

area, the kinetic parameters, , inv invlnA E , can be obtained by,

inv j j invlnA a b E= + (2.22)

where the subscript j refers to the parameters of Eq. (2.21) obtained at different heating rates jβ .

2.4 Model Free Method

Methodically, model free methods can be classified into three categories: differential

isoconversional, integral isoconversional and modulated thermogravimetry methods [52].

2.4.1 Isoconversional Principle

All isoconversional methods originate from the so called “isoconversional principle” that assumes

reaction rates at a given conversion degree are only a function of temperature [53]; this principle

forms the cornerstone of isoconversional analyses [1, 54, 55]. Taking the logarithmic derivative

of the reaction rate of the general kinetic equation, Eq. (1.7), at a given α , one gets

17

( ) ( ) ( )ααα

αα

∂∂

+

∂∂

=

∂∂

−−− 111lnln/ln

Tf

TTk

Tdtd

(2.23)

Note that at a given α , ( )αf remains constant and the second term on the right hand side of the

above equation is zero. Thus,

( )R

ET

dtd α

α

α−=

∂∂

−1

/ln (2.24)

In order to obtain the activation energy, αE , three-to-five experimental tests at different heating

protocols, such as at different heating rates, have to be performed. Of particular importance is that

the procedure for extracting activation energy values does not require any assumption about or

determination of the reaction model. For this reason, isoconversional methods are often called

model-free methods. However, it has to keep in mind that although they do not require the

reaction model to be identified, they do assume that the conversion dependence of the rate follow

the same reaction model of const=α .

A large number of isoconversional methods have been developed. In general, the methodology of

model free methods can be classified into three categories: differential isoconversional [56-58],

integral isoconversional and modulated thermogravimetry methods [52]; integral isoconversional

methods include regular integral methods [59-66] and advanced (or “incremental” in some

studies) integral methods [67-71].

2.4.2 Differential Isoconversional Methods

The differential isoconversional methods start directly from Eq. (1.7). A classic differential

isoconversional method is the Friedman (FR) method [56], which is derived by taking logarithms

of both sides of the general kinetic equation under different heating protocols, iβ ,

18

( )ln ln lnii

d E A fdT RTαβ α = − + +

(2.25)

The index i is used to denote various temperature programs and Ti is the temperature at which

the given conversion degree, α , is reached under the corresponding heating program. Using the

isoconversional principle, the term ( )ln lnA f α+ in Eq. (2.25) remains unchanged for a given

α ; then, a plot of ( )ln /i d dTβ α against 1/ iRT determines the value of Eα . However, the

FR method and other differential isoconversional methods are very sensitive to experimental

noise, resulting in large deviations of Eα , which limits their application in assessing solid state

reactions [54].

2.4.3 Regular Integral Methods

Regular integral methods start from the integral form of the general kinetic equation, Eq. (2.2). In

past decades, a variety of regular integral methods have been developed. This section presents in

detail the methodology of some of the most popular methods.

2.4.3.1 Ozawa-Flynn-Wall Method

The Ozawa-Flynn-Wall (OFW) method [72, 73] starts from Eq. (2.2) and employs the Doyle

approximation [22-24] of ( )p x to yield,

( ) 331.5ln052.1ln −+−=α

βRg

AERTE

ii (2.26)

The value of αE is then evaluated from the slope of the linear plot of iβln against iRT/1 .

Note that the OFW method involves two systematic errors sources. First, it starts from Eq. (2.2),

which involves the assumption that E must remain constant with respect to α , while for complex

19

reactions E varies with α . Second, even if E is a constant, the OFW method still contains the

error sources associated with the Doyle approximation [74] of p(x).

2.4.3.2 Vyazovkin Method

The method [67] proposed by Vyazovkin also starts from the integral form of the general kinetic

equation,

( ) [ ]0 0

exp ,T

ii i i

A E Ad dt A J E Tf RT

ααα α α

α ααα β β

−= =

∫ ∫ (2.27)

It employs a given conversion and a set of experiments performed under n arbitrary heating

programs:

[ ] [ ] [ ]1 21 2

, , , nn

A A AJ E T J E T J E Tα α αα α αβ β β

= = = (2.28)

Numerically, after canceling Aα the value of Eα can be determined by minimizing the following

function:

( )( )1

,,

n ni

i j i j

J E T tJ E T t

α α

α α= ≠

∑∑ (2.29)

Both the model fitting method and IKP method, as well as some other methods such as the

Kissinger method [61], provide only a single pair of E and A while the value of E obtained by an

isoconversional method varies with the progress of conversion degree, α . Vyazovkin

recommended the concept of “variable activation energy” in a review article [75] entitled ‘kinetic

concepts of thermally stimulated reactions in solids: a view from a historical perspective’ but it

has aroused strong controversies [76-78]. For example, Galwey [78] gave critical scrutiny of the

consequences of using the concept to the theory of the subject. It stated that in some systems the

20

initial solid state reactant would have melted before the reaction of interest or the kinetic behavior

pattern would have been adequately explained by contributions from complex or secondary

controls. Therefore, it argued that the supporting information provided in the article [75] was

insufficient, unsatisfactory and unnecessary for introducing the concept of variable activation

energy. It also claimed that, although considerable theoretical problems exist in current

understanding of reactions proceeding in solid phases, the long term development of the subject is

best approached by individually identifying and quantitatively determining the contributing

factors controlling or influencing the rate of any reaction of interest. It concluded that the

introduction of variable activation energy was a retrograde step, unlikely to progress science

through the development of theory, and does not recommend to use. In a short article [79],

Vyazovkin replied [80] that for the condensed phase the free energy of activation did not have to

be the free energy or enthalpy of activation but, rather, a function of temperature dependent

properties of the reaction medium; hence, it claimed that the term of variable activation energy,

which is often called ‘actual, effective, empirical and not merely theoretical’, was a reasonable

compromise between the complexity of solid reactions and oversimplified methods used to

describe their kinetics . Vyazovkin expressed the viewpoint that, by accepting variable activation

energy as a practical compromise, people would forego the methods producing single values of

the activation energy and begin using multiple run methods (such as isoconversional methods)

that allow for detecting reaction complexity [81]. And in recent years, the terms ‘variable

activation energy’ and ‘model free methods’ have become popular and widely used.

2.4.3.3 Li-Tang Method

Li and Tang [64-66] proposed an isoconversional method for the analysis of thermoanalytical

data. It firstly transforms the general equation into the following equation,

( )∫ ∫∫∫ +−=−=

=

α ααα

αααααβαα0 000

lnln GT

dREd

TREd

dTdd

dtd

(2.30)

21

where ( ) ( )[ ]∫+=α

αααα0

lnln dfAG . Because of the isoconversional principle applied in

model free methods, ( )αG would be a constant for several different heating programs for a given

α ; a plot of ( )∫α

αα0

/ln ddtd against ( )∫α

α0

/1 dT would determine the values of activation

energy from the slope of the linear plot. Like the FR method, the Li-Tang (LT) method removes

systematic errors associated with approximating the exponential integral. However, the derivation

of Eq. (2.26) also involves the assumption that E must be a constant with a change in α . Note

that this method is more tolerant of noise than the FR method because the FR method uses the

differential data within plots of ( )[ ]dTdi /ln αβ versus iRT/1 ; in the LT method, the logarithms

followed by the integration decreases the influence of noise.

Budrugeac et al. [82] pointed out that it is difficult to determine initiation temperatures of a

reaction when using the LT method and recommended a similar equation with a non-zero lower

limit of α for integration,

( )∫ ∫ +−=

α

α

α

αααααβ

1 1

ln GT

dREd

dTd

(2.31)

where the lower limit of integral could be 01 >α , and ( ) ( ) ( )[ ]∫+−=α

αααααα

1

lnln1 dfAG .

However, it has been shown that the activation energy obtained by this method depends on the

lower limit of integration, 1α , and the activation energy is missed when 1α α< [82]. Thus,

Budrugeac et al. considered the LT method unsuitable for determining the dependence of E as a

function of conversion degree [82].

2.4.3.4 Kissinger-Akahira-Sunose Method

The Kissinger-Akahira-Sunose (KAS) method [61, 83] employs the Coats-Redfern [25]

22

approximations of p(x) to yield,

( ) ii RTE

EgAR

T−=

αβ lnln 2 (2.32)

The value of αE can be evaluated by plotting the left side of the equation versus iRT/1 .

2.4.4 Advanced Integral Methods

In the derivation of Eq. (2.2) for regular methods, it is assumed that E must be a constant; this

assumption is especially problematic if multiple reaction and/or complex reaction mechanisms

are involved. To avoid this disadvantage, advanced integral methods have been developed that

start from a modified version of the integral form of the general kinetic equation,

( ) ( ) ∫∫∆−

−==∆−

∆−

α

ααβα

αααα

α

αα

T

TdT

RTEAd

fg exp1, (2.33)

where α∆ is a small reaction segment. Since the integration is applied to a small segment of

conversion degree, it is reasonable to take E as a constant.

2.4.4.1 Advanced Vyazovkin Method

The modified Vyazovkin method [67] assumes E to be constant only for a small segment α∆

and uses integration over small time segments for Eq. (2.27),

( ) ( ) ( )1exp ,a

t

ti

Ed A dt A J E T tf RT t

α

α

αα

α α α αα α

αα −∆−∆

−= =

∫ ∫ (2.34)

The procedure utilizes a given conversion and a set of experiments performed under n arbitrary

heating programs:

( ) ( ) ( )1 2, , , nA J E T t A J E T t A J E T tα α α α α α α α α= = = (2.35)

23

Numerically, after canceling Aα , the value of Eα can be determined by minimizing the following

function:

( )( )1

,,

n ni

i j i j

J E T tJ E T t

α α

α α= ≠

∑∑ (3.36)

2.4.4.2 Advanced Li-Tang Method

During the deviation of the LT method or the procedure improved by Budrugeac [82], E and A

should be independent of the conversion degree. Otherwise the integration from 0 or 1α to the

current α will lead to systematic errors. However, systematic errors can be minimized if the

equation is integrated over a very small interval of conversion degree, α∆ , since the activation

energy can be regarded as constant within this very small segment. Thus, Eq. (2.30) can be

converted into,

( )ln d E dd Gdt R T

α α

α α α α

α αα α−∆ −∆

= − + ∫ ∫ (2.37)

where:

( ) ( )ln lnG A f dα

α αα α α α

−∆= ∆ + ∫ (2.38)

in which, α varies from 3 / 2α∆ to 1 / 2α−∆ with a step ( )1/ 1mα∆ = + , and where m is the

number of the equidistant values of α . Plotting the left side of Eq. (2.38) versus the integration

of the reciprocal of the temperature should give a linear plot with Eα obtained from the slope of

the regression line.

24

2.4.5 Modulated Thermogravimetry Methods

Modulated thermogravimetry methods [84-91] use nonlinear heating rate programs during

thermal analyses. Since they are rarely used, this section presents only a brief discussion about

their methodology. For example, the temperature-jump method [91] modulates the temperature

by making it quickly increase from one value to another at a certain moment in time; during this

‘jump’ transition it is assumed that the extent of conversion remains unchanged. The value of Eα

at the given conversion degree is then obtained from a single heating program rather than

multiple ones. In fact, the modulated thermogravimetry method indeed uses the isoconversional

principle, and the test results can be taken as experimental realization of the principle [1]. Other

modulated programs [92, 93] may employ other temperature modulation forms such as

( )0 sin 2T T t L tβ πω= + + , where ω and L are the frequency and amplitude of the modulation.

However, all of them cannot avoid depending upon the isoconversional principle.

25

CHAPTER 3: MODIFICATION OF REGULAR LI-TANG METHOD AND ORTEGA

METHOD

Differential methods have been considered to be potentially more accurate than integral methods

(like the OFW method) because differential methods do not employ any approximations [2].

However, the practical use of differential methods unavoidably involves some inaccuracy: first,

if differential methods are applied to assess differential data from DSC or DTA significant

inaccuracy may be introduced because of the difficulty to accurately determine the data baseline.

Second, experimental noise could lead to significant errors to applying the differential methods

like FR and may also introduce inaccuracies when the raw data are smoothed. A major advantage

of the integral methods is that they avoid these limitations due to the usage of integral data. It

should also be noted that, in the derivation of regular integral methods, E must be independent of

α or otherwise the integration from 0 to α in Eq. (2.2) would result in serious systematic errors

[29, 94]. All regular integration methods [61, 64-66, 72, 73, 95-97] are prone to the same

problems, some of which are also influenced by the temperature integral approximation.

Fortunately, this kind of limitation can be overcome by using the advanced Vyazovkin (AIC)

method [67]. Other recently developed methods [58, 68, 70, 98-100] have made minor

modifications to the AIC method to decrease computational efforts. Overall, the key idea behind

these incremental methods is to calculate Eα within a small segment.

This chapter discusses two examples to make a clear understanding of the above general

comments on model free methods. In the first example, a simple and precise incremental

isoconversional integral method based on Li-Tang (LT) method is proposed for kinetic analysis of

solid thermal decomposition in order to evaluate the activation energy as a function of conversion

degree. This new approach overcomes the limitation of the LT method and eliminates the problem

of the calculated activation energy being influenced by the lower limit of integration. Shown is

the dependence of activation energy on conversion degree evaluated by this new method that is

26

consistent with results obtained by the Friedman (FR) method and the modified Vyazovkin

method for the kinetic analysis of both simulated nonisothermal data and experimental data from

the decomposition of strontium carbonate. Because the new method is free from approximating

the temperature integral and not sensitive to kinetic data noise, it is believed to be more

convenient for assessing nonisothermal kinetic data acquired during solid decomposition.

The second example examines the average linear integral isoconversional method developed by

Ortega and its improvement. Because evaluations of the activation energies of solid state

reactions may be hindered by experimental noise and the uncertainties associated with selecting

appropriate reaction segments, this research suggested a procedure, called the modified Ortega

method, which can avoid or minimize these hindrances. A more consistent dependence of the

activation energy on the extent of reaction conversion was found when using this modified Ortega

method to assess both simulated and experimental data and these results were more in-line with

those calculated using the modified Vyazovkin method and the Friedman method.

3.1 Advanced Li-Tang Method

3.1.1 Methodology of Advanced Li-Tang Method

The integral form of the general kinetic equation is,

( ) ( ) ( )0 0

expTd A E AEg dT p x

f RT Rα αα

α β β = = − = ∫ ∫ (3.1)

where ( )p x is the temperature integral, which has no analytical solution. In the classic integral

isoconversional methods, such as the Ozawa-Flynn-Wall method (OFW) [72, 73], the

approximations of ( )p x [23, 25, 26] should be adopted, the result of which is the introduction of

systematic errors in calculating the activation energy [101].

27

To avoid the usage of the temperature integral approximation, Li and Tang [64, 66, 97] proposed

an isoconversional method for analyzing thermal analytical data. Their method takes the

logarithm and integrates both sides of Eq. (3.1),

( )0 0 0

ln lnd d E dd d Gdt dT R T

α α αα α αα β α α = = − + ∫ ∫ ∫ (3.2)

where

( ) ( )0

ln lnG A f dα

α α α α= + ∫ (3.3)

Because ( )G α is constant for a given α for different heating programs, a plot of

( )0

ln /d dt dα

α α∫ against ( )0

1/ T dα

α∫ will be a straight line and the value of the activation

energy Eα can be obtained from the slope of the line.

In agreement with the Friedman (FR) method [56], the Li-Tang method (LT) avoids systematic

errors introduced by the temperature integral approximations during the calculation of activation

energy. Moreover, this method is more tolerant of data noise in calculating activation energy than

the Friedman method because the data sets of ( )0

ln /d dt dα

α α∫ ~ ( )0

1/ T dα

α∫ of the LT

method are less sensitive to raw data noise than those of ( )ln /d dtα ~1/T of the Friedman

method.

Budrugeac et al [82] pointed out that it is difficult to determine the initiation point of a solid

reaction when using the LT method, and thus recommended an improved version of Eq. (3.2)

with a non-zero lower limit of α for integration,

( )1 1

ln d E dd Gdt R T

α α

α α

α αα α = − + ∫ ∫ (3.4)

28

where the lower limit of the integral is 1 0α > , and

( ) ( ) ( )1

1 ln lnG A f dα

αα α α α α= − + ∫ (3.5)

However, it has been shown that the activation energy obtained by this improved method depends

on the lower limit of integration, 1α , and the activation energy with 1α α< is missed [82]. Thus,

Budrugeac et al. considered that the LT method was not suitable to find the dependence of

( )E E α= [82].

In this dissertation, an incremental version of the LT method, which is independent of the lower

limit of integration, is proposed and verified by numerical and experimental examples. The values

of activation energy calculated by this new method are compared with those obtained by other

isoconversional methods, such as the LT method and its improved version by Budrugeac et al.,

the OFW method, the FR method and the modified Vyazovkin method (AIC) [67]).

In the deviation of the LT method or the procedure improved by Budrugeac et al., E and A should

be independent of the conversion degree. Otherwise, the integration from 0 or 1α to a current α

value will lead to systematic errors. However, the systematic error can be minimized if Eq. (1) is

integrated over a very small interval of conversion degree, α∆ , since the activation energy can

be regarded as constant within a very small segment. Thus Eq. (3.2) can be changed to

( )ln d E dd Gdt R T

α α

α α α α

α αα α−∆ −∆

= − + ∫ ∫ (3.6)

where

( ) ( )ln lnG A f dα

α αα α α α

−∆= ∆ + ∫ (3.7)

29

in which α varies from 3 / 2α∆ to 1 / 2α−∆ with a step ( )1/ 1mα∆ = + , where m is the

number of the equidistant values of α . The plot of the left side of Eq. (3.7) versus the integration

of the reciprocal of temperature should be a linear and Eα can be obtained from the slope of the

regression line.

Obviously, the above incremental isoconversional method avoids the problem of the LT method

having its calculated activation energy dependent on the lower limit of integration. Hence, this

new method is expected to give more consistent results with those from Friedman method (for

noise-free data) or the modified Vyazovkin method.

3.1.2 Numerical Applications

This section will verify the advantage of the new incremental isoconversional method by

numerical examples. Unlike experimental data on solid state reactions, the simulated data are not

affected by noise and therefore are most suitable to test the newly proposed method. To evaluate

the performance of the new method, the FR and AIC methods are also used to analyze the

simulated data and the results are compared. The FR method is chosen because it is directly based

on the general kinetic equation, and thus gives reliable activation energy values for the simulated

data which are not encumbered by noise. Similarly, the AIC method is chosen because it is

believed to be an accurate integral isoconversional method although it is complex to perform

[67]. It is noted that this approach to test the quality of an isoconversional method appears in

many published manuscripts [58, 67, 68, 96, 98, 102].

In this dissertation, a process that involves two parallel reactions and a variation in the effective

activation energy is simulated. The overall kinetic equation of this process is described as:

( ) ( )21 1 2 2exp 1 exp 1A E A EddT RT RTα α α

β β = − − + − −

(3.8)

30

where 1E = 80kJ/mol, 1A = 810 1min− , 2E = 160 kJ/mol, 2A = 1610 1min− ; four linear heating

rates, 1 4β − =1, 2, 4, 8 K/min, are used. Note that equations similar to Eq. (3.8) are used in the

aforementioned papers [58, 67, 68, 96, 98, 102] but with different Arrhenius parameters or model

functions.

The dependence of the apparent activation energy as a function of the conversion degree obtained

by aforementioned isoconversional methods is displayed in Fig. 3.1. It indicates that:

-The Eα dependence calculated by the new method is practically identical to that estimated by

FR method and AIC method, i.e. newE ≈ AICE ≈ FRE ;

Figure 3.1 Eα dependencies evaluated for the simulated process by Li-Tang method

with different lower limit of the integral, αl, as well as by OFW, AIC, FR and the new

method

0.0 0.2 0.4 0.6 0.8 1.060

80

100

120

140

160

E αkJ

/mol

α

new method

FRAIC

OFW

LT LT(α1=0.1)LT(α1=0.2)

LT(α1=0.3)

31

-The Eα dependence estimated by the LT method (Eq. (3.2)) and the version improved by

Budrugeac et al. (Eq. (3.4)) deviates noticeably from the dependence estimated by the FR

method;

-As noted by Budrugeac et al.[82] , the activation energy values obtained by LT method depend

on the lower limit of the integral 1α .With the increases of the lower limit of 1α , the information

of Eα for 1α α< will be lost.

These simulated data suggest that the new method gives reliable activation energy when E varies

with the degree of conversion. It can be concluded that the new method is much better than the

regular LT method and the method improved by Budrugeac.

0.0 0.2 0.4 0.6 0.8 1.080

120

160

200

240

280

E αkJ

/mol

α

FRAIC

new method

Figure 3.2 Eα dependencies evaluated for the SrCO3 decomposition by the new

method, FR method and AIC method

32

3.1.3 Experimental Example

The thermal decomposition of strontium carbonate ( 3SrCO ), which is used as the experimental

example in this dissertation, was carried out in a 50 ml/min flow of 2N with heating rates of 0.5,

5, 7.5 K/min from room temperature (300K) to 1000K on a Shimadzu DTG-60H TGA/DTA

Analyzer. The 3SrCO sample (purity of >99.99%) was supplied by Tianjin Guangfu at the Fine

Chemical Research Institute. The sample was dried for two hours at 450oC and then between

23.8-24.3 mg was loaded into the TGA/DTA sample holder. Although heating rates of 0.5, 5, 7.5

K/min were to be used during the testing, actual heating rate values were calculated from the

recorded sample temperature against time.

0.0 0.2 0.4 0.6 0.8 1.0100

120

140

160

180

200

220

240

E αkJ

/mol

α

new method

LT(α1=0.45)

LT(α1=0.3)

LT(α1=0.15)LTOFW

Figure 3.3 Eα dependencies obtained for the SrCO3 decomposition by the new

method and Li-Tang method with different lower integral limit,αl , and by OFW method

The dependence of activation energy on the conversion degree obtained by aforementioned

isoconversional methods is shown in Fig. 3.2 and Fig. 3.3. From Fig. 3.2, it can be seen that the

activation energy decreases with the increasing conversion degrees. The results obtained by the

33

new method and AIC method are consistent, while those obtained using the FR method seem less

consistent, perhaps because of effects of experimental noise. Taking the values calculated by the

Modified Vyazovkin method as a reference, the overall standard deviation of the E values

determined by new method was 4.14 while the overall standard deviation of the values calculated

by FR method was 17.79. Moreover, from Fig. 3.3 it can be seen that:

-the OFW method gave much lower values near the onset ( 0.15α < ) of the reaction, and then

rather stable values for 0.15α > .

-the LT method gave consistent values in the beginning of the process but also led to rather stable

values for 0.15α > . As expected, the dependence of Eα values obtained by the Budrugeac et al.

method depended on the lower limit 1α in Eq. (3.4).

-The values of the activation energy obtained by the OFW method, LT method, and the method

improved by Budrugeacet et al. differed considerably from the results obtained by AIC (Fig. 3.2).

From Figs. 3.2 and 3.3 it can be shown that the proposed new approach has distinct advantages

over some integral isoconversional methods (OFW, LT etc.). It can be concluded that the new

approach is capable of providing consistent values of activation energy obtained by AIC even if E

varies strongly with the conversion degree.

3.2 Modified Ortega Method

3.2.1 Methodology of Modified Ortega Method

Recently, Ortega developed a simple average integral isoconversional method [98] (the original

Ortega method) for the most frequently used linear heating program, ( ) /dT t dtβ = , of

nonisothermal experiments , which is based on the integral form of the general kinetic equation,

( ) ( )0

1 expt

o

Eg d A dtf RT

αα α

α = = − ∫ ∫ (3.9)

34

It leads to

( ) ( )0 0exp

Td A Eg dTf RT

α ααα β

− = = ∫ ∫ (3.10)

For a small segment α∆ , Eq. (3.10) can be approximated by

( ), exp expa

T

T

E EA A Tg dTRT RT

α

α

α α

α

α α αβ β−∆

− ∆ − ∆ = ≈ −

∫ (3.11)

where T T Tα α α−∆∆ = − .

Taking the log of Eq. (3.11) yields

( )ln , ln ln ETg ART

α

α

α α αβ

∆−∆ ≈ + −

(3.12)

For a given conversion and set of n experiments carried out at different linear heating programs

iβ ( i =1, …, n ) , Eq. (3.12) leads to

, ,

ln .i

i i

EconsT RT

α

α α

β = − ∆

(3.13)

Then a plot of ( ),ln /i iTαβ ∆ versus ,1/ iRTα will be a straight line and the value of Eα can be

determined from the slope of the line.

Using a traditional integral isoconversional method, such as the Ozawa-Flynn-Wall (OFW)

method [72, 73], creates system errors when the value of E varies with the conversion degree α .

To avoid these errors, the Ortega method and some other newly developed methods [67, 68, 70]

also use small integral segments, α∆ . However, when α∆ is small, the use of a single

temperature Tα prevents accurate characterization of the reaction segment and creates a source of

systematic error. Hence, the Ortega method requires accurate values of Tα α−∆ and Tα , and

potentially large α∆ , to minimize the influence of temperature measurement noise; errors in the

temperature interval T∆ compromise the accuracy of Eα values. The accuracy of the improved

35

version of the Ortega method is examined using both numerical and experimental examples, and

some recommendations for using this method are discussed in the following.

The derivation of Eq. (3.11) imposes the integration range from α α−∆ to α . Because of the

importance of selecting an appropriate temperature to characterize the temperature segment when

using the modified Ortega method, the integration of Eq. (3.11) with respect to α is changed

between / 2α α−∆ and / 2α α+ ∆ , which yields

( ) /2

/2

/ 2, / 2 exp expT

T

E EA Ag dT TRT RT

α α

α α

α α

ε

α α α αβ β

+∆

−∆

− − ∆ + ∆ = = − ∆

∫ (3.14)

where /2 /2T T Tα α α α+ −∆ = − , /2 /2nT T Tα α ε α α−∆ +∆≤ ≤ .

As the second step, a number, n, of temperatures, nTε , are selected to characterize the temperature

for different reaction segments; hence, nTε varies from /2Tα α−∆ to /2Tα α+∆ with the step

( )/ 1h nα= ∆ − and n an odd number, as shown below:

if 1n = , 1T Tε α=

if 3n = , 1 /2T Tε α −∆= , 2T Tε α= , 3 /2T Tε α +∆=

if 5n = , 1 /2T Tε α −∆= , 2 /4T Tε α −∆= , 3T Tε α= , 4 /4T Tε α +∆= , 5 /2T Tε α +∆=

…

Finally, the characteristic temperature of the aforementioned n temperatures is defined as

1

1 n

ii

T Tnε ε

=

= ∑ (3.15)

36

Thereby, noise created by temperature measurements when using small α∆ can be minimized

because a relatively large reaction segment α∆ is used for a large number n temperatures. For a

set of experiments carried out at different heating rates, Eq. (3.13) can be modified to

,

ln .i

i

EconsT RT

α

α ε

β = − ∆

(3.16)

The activation energy, Eα , is obtained from a plot of ( ),ln /i iTαβ ∆ against 1/ RTε .

Figure 3.4 Dependence of Eα on the value α evaluated by FR, AIC, OFW, the original

Ortega method, and the modified Ortega method with ∆α=0.04

It is proposed that this modified Ortega method can minimize the systematic error caused by large

α∆ segments and increase tolerance to experimental noise as n is increased. In the following

section, the modified Ortega method [71] is assessed for the case of n = 5.

0.0 0.2 0.4 0.6 0.8 1.072

75

78

81

84

87

FR method New method AIC method Ortega method OFW method

α

E αkJ

/mol

37

3.2.2 Results and Discussion

Both simulated and experimental data were assessed to validate the modified Ortega method. The

first simulation procedure described below used two parallel reactions without noise: one was a

first order reaction and the other a second order reaction. In the second simulation discussed

below, a single step reaction with noise was assessed to explore effects of noise on the calculated

Eα values. As for applicability of the modified Ortega method to experimental data, data during

the decomposition of strontium carbonate ( 3SrCO ) were analyzed using the modified Ortega and

other methods.

3.2.2.1 Simulation without Noise

The modified Ortega method was compared to the original Ortega method, and the Friedman

(FR) and modified Vyazovkin (AIC) methods, when using simulated data. The FR method is

known to be very sensitive to experimental noise that can lead to serious deviations, but without

noise it can give reliable activation energies. Similarly, the AIC method is known to be one of the

most accurate integral isoconversional methods with good tolerance of noise although it is more

complex to perform. Because it was considered beneficial to compare results when using the

modified Ortega method and the Ortega, FR and AIC methods with a more traditional integral

method, the OFW method was also used to calculate activation energies.

For a process having two parallel reactions, the activation energy varies with α and the overall

kinetic equation is described as

( ) ( )21 1 2 2exp 1 exp 1A E A EddT RT RTα α α

β β = − − + − −

(3.17)

where 1E =100 kJ/mol, 1A = 910 1min− , 2E =75 kJ/mol, 2A = 810 1min− ; four linear heating rates

of 1 4β − = 2, 4, 6, 8 K/min were studied.

38

Figure 4.4 compares the dependence of calculated Eα values as a function of α for the five

different isoconversional methods when an integral segment α∆ = 0.04. The following

summarizes the findings:

• The Eα dependence determined by the modified Ortega method leads to practically the

same result estimated by FR and AIC methods;

• The Eα dependence calculated by the original Ortega method had larger systematic

errors as compared to its dependence calculated by the FR method; and

• The Eα dependence obtained by OFW method differed noticeably from the results

calculated by other methods, because traditional isconversional methods that assume E is

independent of α lead to a large systematic error when E varies with α [16, 17].

0.0 0.2 0.4 0.6 0.8 1.085

90

95

100

105

110

α

E αkJ

/mol

Preset value FR method AIC method New method Ortega method

Figure 3.5 Dependence of Eα on the value α for the simulated single step reaction

FR, AIC, and the original Ortega method with ∆α =0.01, and the modified Ortega

method with ∆α =0.04

39

3.2.2.2 Simulation with Noise

To evaluate the effect of noise on the value of Eα , a small-order, random temperature noise

(−0.01 to 0.01 K) was introduced into the simulated data when using a single step reaction:

( )exp 1d A EdT RTα α

β = − −

(3.18)

where E=100 kJ/mol (a setting value), A = 910 and 1 4β − = 2, 4, 6, 8 K/min.

0.0 0.2 0.4 0.6 0.8 1.080

120

160

200

240

280

α

E αkJ

/mol

AIC method New method Ortega method


decomposition by AIC, the modified Ortega method, and the original Ortega method, all

with ∆α =0.04

Figure 3.5 compares the results, and the following summarizes them:

-The Eα dependence estimated by the modified Ortega method ( α∆ = 0.04 in this simulation)

gave results consistent with the AIC method if relatively large segments α∆ were used;

40

-The Eα dependence calculated by both the original Ortega method ( α∆ = 0.01) and FR method

led to noticeable deviations;

-Also, the original Ortega method led to larger deviations as smaller Δα segments were used.

3.2.2.3 Experimental Application

The thermal decomposition of 3SrCO [70] was examined again, using a 2N flow rate of 50

ml/min and three different heating rates: 2.379, 4,734, and 7.007 K 1min− ; the data were

acquired using a Shimadzu DTG-60H simultaneous TGA/DTA analyzer. The results from

analyzing the data using the five different methods were then assessed to check on the accuracy

of the dependence of Eα on the value of α .

0.0 0.2 0.4 0.6 0.8 1.080

120

160

200

240

280

α

E αkJ

/mol

AIC method FR method Ortega method


decomposition by AIC, FR and Ortega method all with ∆α=0.04

41

For α∆ = 0.04, Fig. 3.6 shows that the Eα dependence calculated by the modified Ortega

method and AIC methods agreed very well, while the original Ortega method data deviated from

the modified Ortega method and AIC data due to the expected influence of systematic error and

experimental noise. Taking the values obtained by the AIC method as the reference benchmark,

the standard deviation (SD) of Eα calculated by the modified Ortega method was SD = 0.53; in

contrast, the original Ortega method resulted in SD = 17.14, more than 30 times larger. Figure 3.7

shows that both FR and OFW methods largely disagreed with the benchmark AIC method.

3.3 Conclusions

The original Ortega’s average integral isoconversional method only uses the upper temperature

limit to characterize selected reaction segments to evaluate the activation energy for solid state

reactions. In this study, a modified procedure based on the original method was developed

without any additional assumptions. The modified method chose larger reaction segments and

several temperature values rather than a single temperature to characterize the integral segment.

Using the modified Ortega method not only eliminated systematic errors but also effectively

reduced the influence of experimental noise. The validity of the modified Ortega method was

shown by both simulated and experimental data.

During the development of the modified Ortega method, it was found that certain specifications

for the analyses were important in providing the best results, and did not impart unwanted errors

or misrepresentations of the actual data. These specifications included: (1) The segment of

temperature to characterize temperature corresponding to α should not be too small, i.e.

Δα ≥ 0.04, and (2) the number of characteristic temperatures used should be n ≥ 5.

An incremental isoconversional method has been developed based on Li-Tang method without

any additional assumptions. The incremental version not only avoided the integration of the rate

equation and lowered the effects of noise which are encountered in Friedman method, but also

42

eradicated the limitation of Li-Tang method that the activation energy would depend on lower

limit of the integration. Moreover, the procedure was very simple and gave consistent activation

energy values with those obtained by Friedman and the modified Vyazovkin methods.

However, it should keep in mind that original LT, Ortega methods, the modified version of them,

and all other isoconversional methods, are based on the isoconversional principle, the influences

of which are not taken into consideration in the analyses of this chapter. Therefore, in next

chapter, additional examinations of the conclusions in this chapter are given and a comprehensive

method is proposed which is able to take advantage and avoid the limitations of these methods.

43

CHAPTER 4: COMPREHENSIVE METHOD BASED ON MODEL-FREE AND IKP

METHODS TO EVALUATE KINETIC PARAMETERS

Evaluating Eα without any previous knowledge of ( )f α is considered a substantial advantage

of using model free methods. Moreover, having an variable activation energy, Eα , that varies as

a function of conversion degree, α , is believed beneficial for revealing the complexities inherent

to solid-state kinetics [35, 36]. However, the phenomenon of E varying with α and the

inconsistencies in results obtained during various studies [17] have caused debate and controversy

[103, 104]. Explanations of these inconsistencies have often focused on the complexities

associated with solid state experiments [105-107] and the introduction of systematic errors

associated with computational methods, one of which is the approximation made during

integration, as shown in Eq. (2.2) [108]. To perform model free analysis accurately, great care

should be taken to ensure that each experiment is performed at the same conditions with constant

mass and size of samples, and constant gas purge rate, etc. Variations in experimental conditions

can be minimized and systematic errors associated with computational methods can be eliminated

by using incremental approaches such as the AIC and the MLT methods. However, although

some studies [11] showed that the results obtained by isoconversional methods depend on the

heating rate, few in-depth assessments of the validity of the isoconversional principle have been

published even though it is the foundation of model free methods - as embodied by Eq. (2.24). To

begin such an assessment, this study used a linear heating program to provide critical information

for assessing the isoconversional principle. More importantly, a comprehensive method is

proposed that extracts more meaningful and reliable kinetic parameters of solid state reactions.

The structure of this chapter is set as follows. In Section 4.1, a theoretical approach is suggested

that is used to assess whether the “isoconversional principle” of Eq. (2.24) is valid for single step

and complex reactions. In Section 4.2, the traditional IKP method is modified to simultaneously

determine variable activation energies and pre-exponential factors for complex reactions. In

44

Section 4.3 a comprehensive method is proposed that is based on both isoconversional and IKP

methods.

4.1 Critical Analysis of Model Free Methods

The overall kinetic equations of parallel independent reactions are given as,

( ) ( ) ( ) ( )1 2 1 1 2 21 1 2 21 exp 1 expd d A E A Ed c c c f c f

dT dT dT RT RTα αα α α

β β = + − = − + − −

(4.1)

where ( )0,1c∈ is the contribution percentage of the first reaction to the change in the overall

reaction, ( )1 21c cα α α= + − . Then, the “apparent”, “overall”, “empirical” or “global” [54]

activation energy for expressing the activation energy at a given α is,

( )

( )

( )1 2 1 21 2 1 2

1 1 2 21 2

1 1

1

d d d dc E c E c E c EdT dT dT dTE f E f Ed d dc c

dTdT dT

α

α α α α

α α α

+ − + −= = = +

+ − (4.2)

where 1f and 2f are contribution percent of the overall reaction rate /d dTα , and 1 2 1f f+ = .

It is expected that, for a given conversion degree, the value of 1f and 2f will vary with heating

rate; consequently, E is not only a function of α but also a function of β , as is shown

explicitly in the following simulations.

Consider a simple case in which the values of ( )Af α for two reactions are the same, i.e.,

( ) ( )1 21 exp 1 expE Ed c c AfdT RT RTα α

β = − + − −

(4.3)

45

Because temperatures vary with β for a given α , the ratio of ( )1exp /c E RT− -to-

( ) ( )21 exp /c E RT− − of Eq. (4.3) will vary and then the ratio of 1f -to- 2f will also change

with β . Therefore, Eα is expected to also vary with β , that is

( ),appE E α β= (4.4)

where appE is defined as the overall or apparent energy for the complex reaction at given α and

β . Similarly, it is reasonable to express the pre-exponential factor as

( ),appA A α β= (4.5)

Therefore, the starting point of model free methods in which ( )appE E α= is problematic, which

implies the isoconversional principle embodied by Eq. (2.24) cannot be used without taking into

considerations the heating programs.

Similar analyses are possible for other types of complex reactions, such as parallel competitive

reactions:

( ) ( )1 1 2 21 2exp expA E A Ed f f

dT RT RTα α α

β β = − + −

(4.6)

and reversible reactions [109], M N↔ :

( ) ( )1 2exp expN N NM M MM N

d A Ed A E f fdT dT RT RT

αα α αβ β

= − = − − −

(4.7)

where Mα and Nα are the corresponding conversion degrees of substance M and N . In other

words, for complex reactions the dependence of Eα on β has to be tested if the method that

gives activation energies as a function of conversion degree.

46

Figure 4.1 The dependences of E and lnA on α of a single step reaction determined by

IIKP method

4.2 Analysis of IKP Method

Eqs. (2.21) and (2.22) lead to,

2

1invinv inv

max max

ElnA ERT RTβ

= + (4.8)

where the subscript “ max ” means the maximum reaction rate, and “ inv ” refers to invariant

parameters, which historically have been calculated by applying a model-fitting method to a full

reaction range, that is, ( )0,1α ∈ , in a single experimental run. The compensation relations still

hold if a model-fitting method is applied to a small reaction segment α∆ , i.e., from α α−∆ to

α : the characteristic temperature can be approximated as the mean temperature of the selected

computation segment. Therefore, the incremental IKP (IIKP) method has a wider applicability

19

19.5

20

20.5

21

90

95

100

105

110

0 0.2 0.4 0.6 0.8 1

lnA

1/m

in

E k

J/m

ol

α

47

than the IKP method, the possibility of which is seen by considering a simple first order reaction

described by:

( )exp 1 d A EdT RTα α

β = − −

(4.9)

where E = 100 kJ/mol, lnA = 20.7 and 1 4β − = 5, 10, 15, 20 K/min. The values obtained by using

the IIKP method are shown in Fig. 4.1 for an increment of 0.1 and computation range

0.05α∆ = for each point. It can be seen that the values of E and lnA are quite reliable at all

points. Applications of the IIKP method to complex reactions will be discussed in the following

section.

4.3 Proposition of a Comprehensive Method

A schematic for applying the IIKP principle to complex reactions at three different heating rates

is shown in Fig. 4.2. Considering the parallel independent reactions represented by Eq. (4.1), for

a given heating program, β , the temperature of the two reactions is the same at any overall

conversion degree, ( )1 21c cα α α= + − . If the two reactions are assumed to be decoupled, then

each reaction would have individual invariant points, ( )1 1, lnA E and ( )2 2, lnA E , as is shown in

Fig. 4.2. For a given iβ , the slopes of the compensation lines for two reactions would be

identical, i.e,

1 21

i i ib b bRTα

= = = , 1, 2,3i = (4.10)

Therefore, the equations for two reactions can be described as:

for reaction 1-

48

1 1 1 11 1

1 2 1 21 2

1 3 1 31 3

, , ,

lnA b E a forlnA b E a forlnA b E a for

βββ

= += += +

(4.11)

for reaction 2-

2 1 2 12 1

2 2 2 22 2

2 3 2 32 3

, , ,

lnA b E a forlnA b E a forlnA b E a for

βββ

= += += +

(4.12)

where 1b , 2b , 3b , 11a , 21 a … are constants.

Figure 4.2 Schematic of IIKP principle for complex reactions

Note the coefficients of E for a specific β in Eqs. (4.11) and (4.12) are the same. Taking 1β as

an example, the two lines from Eqs. (4.11) and (4.12) are parallel; the combination line is

determined by the contribution ratio, 11 12:f f , of their intercept values, which leads to a

combination of 11 11 12 12f a f a+ . The overall Eq. (4.1) for a given iβ produces the following:

∆E

E1 Einv

lnA1

lnA2

E

P

O

E2

lnAinv

lnA β1

β2

β3

49

1 11 11 12 12 1

2 21 21 22 22 2

3 31 31 32 32 3

, , ,

lnA b E f a f a forlnA b E f a f a forlnA b E f a f a for

βββ

= + += + += + +

(4.13)

From Eq. (4.2), if the contribution percentage remains unchanged, 1 2:i if f m= , for different iβ ,

or if the contribution percentage changes within a narrow range the three lines determined by Eq.

(4.13) intersect at a common point P ( ( ) ( )1 2 / 1mE E m+ + , ( ) ( )1 2 / 1mlnA lnA m+ + ) (see Fig.

4.2) or within a narrow range (Case 1). Otherwise, the three lines intersect in a wider range, E∆ ,

or have no obvious common intersection (Case 2). Note that the “apparent” or “overall”

activation energy appE is determined by the contributions percentage of two individual reactions,

and the activation energy at given α for the Case 1 is independent or weakly dependent on β

whereas for Case 2 it is strongly depend on β .

Taking the log of the general kinetic equation, Eq. (1.7), and differentiating over 1/ T , and then

rearranging it yields,

( )1 1 1

dln lnfE lnAdTT R T T

α

αβ α− − −

∂ ∂∂ = − + +∂ ∂ ∂

(4.14)

If a common point P or narrow range of it exists, then lnA has no or only a weak dependence on

T , in which case the term 1/ 0lnA T −∂ ∂ = . Moreover, considering the assumption of

isoconversional methods that the reaction model remains unchanged for different heating rates,

the last term in Eq. (4.14), ( ) 1/lnf Tα −∂ ∂ , can also be zero. Because the compensation

equation, Eq. (2.21), does not contain ( )f α , the phenomenon of the compensation effect would

determine that the reaction models would have no, or weak, dependence on heating rate.

Therefore, for Case 1, it is reasonable that the activation energy at a given α has no or weak

50

dependence on heating rates and the isoconversional principle gives reliable Eα . These

conclusions suggest the IIKP principle can be used to judge the reliability of the values.

Figure 4.3 Flow chart of comprehensive method

Thereby, it is clear that the new comprehensive method based on model free and IIKP methods

can be proposed, the relationship of which with previous methods are shown in the flow chart

above, in which the blue texts indicates the contributions by the author. The chart shows that the

combination of model free method and IIKP method generates a comprehensive method [110],

which has obvious advantages over previous methods. The steps of the comprehensive method

are described below:

1. Use an incremental model free method to determine values of Eα as a function of α ;

2. Use the IIKP method to calculate both Eα and Aα and check whether the compensation lines at

different heating rates intersect at a common point or within a narrow range. If so, the

Comprehensive method Do not need f(α) Applicable to complex reactions Test the reliability of basic assumption Much more reliable

Model free method Involved with basic assumption Give only E value

Incremental IKP method (IIKP) Applicable to complex reaction

Model fitting

IKP method Applicable to single step reaction

Use data of ∆α ~ ∆T

Use all data of α ~ T

51

corresponding value calculated in step 1 is reliable and can be selected, otherwise the value will

be discarded according to IIKP principle;

3. Use the compensation effect as evidenced by the values determined in step 2 to predict the

values of Aα for the selected values of Eα (this step is discussed more in the simulation section).

It is suggested that the comprehensive method can be applied successfully to other types of

complex reactions, such as parallel dependent reactions. Moreover, it is worth mentioning that the

IIKP principle can be used for arbitrary heating program experiments by replacing /d dTβ α

with /d dtα in the related equations.

Figure 4.4 Values of activation energy determined by FR and IIKP methods for

simulation test S1(The meaning of beta1-4 is the true values calculated by Eq. (4.2) for

β1-4; it applies to Figs. 4.4-4.9)

52


simulation test S2


simulation test S3

53

Figure 4.7 Values of activation energy determined by FR and IIKP methods for simulation test S4

Figure 4.8 Values of activation energy determined by FR and IIKP methods for simulation test S5

54


simulation test S6

4.4 Simulation Validations

A parallel independent reaction was simulated that involved two different reactions, as given by

the following:

( ) ( ) ( )2222

111 1exp11exp α

βα

βα

−

−−+−

−=

RTEAc

RTEAc

dTd

(4.15)

where 1 4β − = 5, 10, 15, 20 K/min, 1lnA = 18.42 1min− , 1E =100 kJ/mol, 2lnA = 36.84 1min− , 2E

= 180 kJ/mol, and the contributions c = 0.1, 0.3, 0.5, 0.7, 0.9 for tests S1-S5 respectively. The

reactions involving S1-S5 may be heavily or totally overlapped. A test S6 with no heavily

overlapped reaction was designed with 2lnA = 28.78.

55

The test S7 simulated a competitive parallel reaction:

( ) ( )αβ

αβ

α−

−+−

−= 1exp1exp 22211

RTEA

RTEA

dTd

(4.16)

where the values of the parameters were identical to those for S1-S5 except 2lnA = 35.69.

The invariant values of E and lnA determined by the IIKP method are shown in Table 2.

To enable reliable comparisons, the isoconversional method used for calculation was the FR

method. Among the isoconversional methods, it is directly derived from the kinetic equation of

Eq. (1.7) [52], and can give the most reliable values of Eα when Eα is not dependent on β .

The values of Eα shown in Figs. 4.4-4.10 were calculated using Eq. (4.2) for every heating rate

along with the values of Eα calculated by the FR method and IIKP method (only the most

frequently used models were applied, noted as ‘*’ in Table 4.1). The major results from the S1-S7

tests are presented below:

1. The apparent activation energy values were dependent on the heating rates. If a dominant

reaction existed - for example for S1 where c = 0.1 or S5 where c = 0.9, the dependence of Eα on

β was weak; if no dominant reaction existed, this dependence was strong.

2. The use of an isoconversional method cannot recognize dependencies of Eα on β , and it can

lead to serious errors even when a dominant reaction exists.

3. The values determined by the IIKP method and the isoconversional method were directly

correlated, and accordingly to the relation of the compensation lines, three cases are presented

below:

56

Case 1: A common intersection point (P in Fig. 4.2) or a narrow intersection domain was found

for which E∆ < 5kJ (empirically). The values of Eα did not depend or weakly depended on β ;

hence, Eα values calculated by the isoconversional method were reliable;

Case 2: A relatively large range (noted as ‘*’ in Table 4.1) in the intersection domain was found.

The values of Eα may have a dependency on β ; the values obtained by the isoconversional

method may be acceptable;


simulation test S7

Case 3: The compensation lines were widely separated or were randomly intersected over a wide

range, E∆ ≥ 20kJ for example. Hence, Eα was strongly dependent on β ; hence, the values

obtained were not reliable – and are not presented herein.

57

4. From the previous discussion, it can be recognized that when results are obtained that reflect

the Case 3 situation, and in many cases the Case 2 situation, the starting point of estimating

activation energy on the given conversion degree is problematic for any model free method,

independent of whether linear heating rates or nonlinear heating programs were implemented.

Table 4.1 The values of E_inv and A_inv obtained by IIKP method

𝛼 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

S1 E lnA

179.1 36.05

177.3 35.65

176.2 35.4

175.4 35.22

174.9 35.13

175.1 35.21

177.3 35.77

188.8* 38.42 ____

S2 E lnA

176.8 35.27

169.8 33.78

165.2 32.76

160.9 31.81

157.5 31.07

158.2* 31.26

188.0* 37.85 ____ 100.0

19.03

S3 E lnA

173.2 34.14

156.7 30.68

144.2 27.97

130.5* 25

115.6* 21.75

114.0* 21.5 ____ 100.0

18.31 100.0 18.58

S4 E lnA

161.5 31.20

131.0 24.88

99.0* 18.12

60.0* 9.85

48.0* 7.08

77.0* 13.02

100.0 17.68

100.0 17.94

100.0 18.18

S5 E lnA

131.4 24.35

84.5* 14.75

52.0* 7.95

88.0* 14.64

105.0 18.05

100.0 17.23

100.0 17.43

100.0 17.69

100.0 17.93

S6 E lnA

101.5 16.31

106.5 16.02

134 19.42 ____ 212*

33.35 181* 28.06

181.5 28.7

181.9 28.78

187.6 29.8

S7 E lnA

178.0* 34.53

179.0 34.98

174.0 34.14

168.0 32.99

165.0 32.38

163.0 31.93

164.0 32.05

168.0 33.77

173.0 33.67

5. When the reactions were partially separated, such as for S1 and S5, or when the reactions were

insignificantly overlapped, such as for S6, the E and A values are considered reliable at the

beginning and/or ending points for both methods. Also, it can be expected that the IIKP method

can be used to extract representative single pairs of E and A for similarly well-separated

reactions. For S7, the trends for the competitive reactions were the same as for S1-S5.

6. The Eα values as determined by the IIKP method deviated from true values to a much greater

degree than by the model free method when a dependence of Eα on β existed. This situation

occurred because the slopes of the compensation lines (the coefficients b in Eqs. (2.21), (2.22)

and (4.8)) were very close. When the heating rate increased from 5 K/min to 40 K/min (most

experimental range), the temperature gap, mT∆ , generated for a given α was generally about

40K, while the characteristic temperature mT tended to be more than 600K. In this situation, the

58

order of the change of coefficient b was much smaller than that of a . For example, in the

simulated data of Eq. (4.9) when the heating rate increased from 5 K/min to 20 K/min the

temperature at α = 0.5 increased about 30 K (533 to 565 K). These variation ranges can be

estimated from:

( )4 41 2.257 10 ,2.129 10bRT

− −= ∈ × ×

( )2 3.162, 3.279Ea lnRT

= ∈ − −

It is to be realized that even a small change in the relative contribution percentage, 1 2:i if f in

Eqs. (4.11) and (4.12), significantly influenced 1 1 2 2i i i if a f a+ in Eq. (4.13) as compared to the

much smaller differences of the slopes, ib ; the result of this influence was that the calculated invE

and invlnA values were noticeably deviated from true values. In general, calculations using

isoconversional methods are based on regression analysis such as least square approaches, and the

results are less influenced by changes in 1 2:i if f than is the IIKP method. For this reason, the

IIKP principle was used to judge the reliability of values obtained by the model free method

rather than directly selecting the values extracted by the IIKP method.

Moreover, it was found that the values of invlnA and invE obtained by using the IIKP method at

different conversion rates were linearly correlated. It is probable this linear relation explains the

existence of single pair values of ( ,inv invlnA E ) that can be determined by using the IKP method.

Because of linearity, it is possible to predict values of lnAα for the selected Eα values. As an

example, if the S3 test was selected for determining lnAα , the conversion degrees selected would

be α = 0.1, 0.2, 0.3, 0.8, 0.9, and the corresponding Eα = 157.89, 159.54, 156.98, 99.99, 100.01

kJ/mol. By using the linear relation and interpolation analysis, the values of pre-exponential

59

factor are then: lnAα = 30.94, 31.29, 30.74, 18.31, 18.31 1min− . Similar calculations could be

accomplished for the other tests.

4.5 Experimental Validation

The thermal decomposition of high purity (> 99%) calcium carbonate ( 3CaCO ) was carried out

in a 40 ml/min flow of 2N using 4.97, 9.92, 14.85, 19.83 K/min linear heating program from

room temperature to 1150K on a NETZSCH STA 409C Analyzer. To minimize variability

between the experiments, the sample was carefully weighted to be between 7.750-to-7.890 mg

before decomposition testing at the different heating rates.

To apply the comprehensive method, two incremental isoconversional methods (AIC, MLT) and

the FR method were applied to analyze the experimental data. Considering that the selection and

use of a reaction model can influence the accuracy of the IIKA method, all the reaction models

listed in Table 1.1 were applied to the experimental data. Data from models having poor fitting

(coefficient < 0.97) were discarded, and the remaining data were selected for determining E

values and then A values with the IIKA method. It can be seen in Fig. 4.11 that:

1. The AIC and MLT methods produced identical Eα dependencies so their curves were merged

in the figure, while the FR method resulted in certain deviations from the AIC and MLT methods

because the FR method is more sensitive to experimental noise;

2. As expected, the Eα dependencies calculated using the IIKA method deviated more seriously

than the AIC and MLT methods. According to the intersection area of the four regression lines

obtained at a given conversion degree, the values determined by the IIKA method were divided

into three levels. The first level was four compensation lines for a given point that intersected

over a narrow range (< 10 kJ/mol) for which the Eα values calculated by the isoconversional

methods were reliable; the second level was four lines that intersected over a range of 10 to 20

60

Figure 4.11 Values of activation energy obtained by different methods for experiment

data

Figure 4.12 Values of selected activation energy determined by comprehensive method

61

kJ/mol (noted with ‘*’), for which the values calculated by isoconversional methods may not

be reliable; and, the third level was four lines that intersected over a range greater than 20kJ or

did not have a common area, the results were not reliable (not displayed).

3. The lnAα values calculated by the IIKA method varied with α and provided a reasonable

compensation relation with Eα that gave 0.1126 1.5944lnA Eα α= − .

In this study, the range of Eα values as determined by the AIC and MLT methods was

175 Eα≤ ≤ 200 kJ/mol; these values are in agreement with previous studies [111, 112] of the

decomposition of 3CaCO in which the range was 170 210Eα≤ ≤ kJ/mol. However, as

discussed herein, these values are considered problematic because they include unreliable values.

Only by applying the comprehensive method proposed in this study is it possible to obtain the

reliable Eα values displayed in Fig. 4.12 with a range of 191.15 194.91Eα≤ ≤ kJ/mol. Using the

relation determined for lnAα , 0.1126 1.5944lnA Eα α= − , that was calculated by the IIKA

method and for which Eα varied with α with a reasonable compensation relation, the values of

lnAα for the selected Eα at corresponding reaction degrees were 119.93 20.36lnA min−≤ ≤ .

These reliable Eα and lnAα values encompass a very narrow range and can be reasonably

approximated as constants with E =192.5 kJ/mol and 120.15lnA min−= .

4.6 Conclusion

Many methods that have been developed to obtain activation energies of solid-state reactions

consider E to be a function only of α . However, after a theoretical study along with examples

presented herein, it is proposed that for complex reactions the values of E is not only a function

of α but also a function of the heating programs. This dependency needs to be considered when

methods are used to obtain Eα dependencies.

Eα

62

It is generally believed that the IKA method requires more computation than other methods and

yet provides only single pair of E and A ; as a consequence, the IKA method is rarely used in

kinetic studies. However, it has been determined that the potential applications and benefits of

this method are underestimated. This research has shown the development and use of the IIKA

method that is based on the original IKA method and gives variable Eα and Aα values for solid

state reactions. It can provide reliable values of Eα and Aα at any given reaction degree for

single-step reactions for simulation data. For experimental data that can be complex reactions and

have a variety of influencing factors such as experimental noise, sample weight and sample size

IIKA can be used to successfully determine the reliability of values obtained by model free

methods, after which a comprehensive method based on IKP and method free methods is

proposed.

This comprehensive method was tested on both simulation and experimental data of the

decomposition of 3CaCO and provided results showing noticeable advantages over other

methods. It can: evaluate the reliability of the results calculated by model free methods;

determine the dependence of E on the heating programs used; select reliable E and provide

variable A values for complex reactions; and to a certain degree, help to judge the quality of the

experimental data.

63

CHAPTER 5: ACCURACY OF ISOCONVERSIONAL METHODS WITH A

CONSIDERATION OF BASIC ASSUMPTION

The comprehensive method proposed in Chapter 4 has many advantages over other existing

methods including isoconversional methods. However, it takes time for people to accept a newly

developed method especially when the current isoconversional methods are widely applied in

almost every field that thermal kinetic analysis is used. In this chapter results are given to

compare the accuracy of existing isoconversional methods by considering the influence of the

isoconversional principle, which is often called the basic assumption. This discussion is helpful

for researchers to select a better isoconversional method that matches with the incremental

invariant kinetic parameters method for different types of complex reactions.

5.1 Previous Comments about Isoconversional Methods

A conclusion of the ICTAC Kinetic Project [32] was that the kinetic analysis of heterogeneous

reactions should use an isoconversional method because it is able to evaluate the activation

energy without any prior knowledge of reaction model. For complex reactions E varies with α .

Studies [29, 94, 113, 114] examining the accuracy of the different types of isoconversional

methods conclude that:

1. the differential methods [56, 58] are very sensitive to experimental noise, resulting in large

deviations of activation energy [54]; for simulation data that do not contain experimental noise,

these methods allows the evaluation of the exact value of activation energy;

2. the regular integral methods [61, 64, 72, 73, 83, 95-97] have good tolerance of noise but,

compared to the “exact” values obtained by the differential methods, they lead to significant

systematic errors because they involve the use of approximations of the temperature integral and

assume E is a constant in the integration of the Eq. (3.2) [94];

64

3. the advanced integral methods [67, 68, 70, 98, 99, 115] that use small integration ranges of the

variables not only have good tolerance of experimental noise but also produce values in

agreement with those obtained by the use of differential methods.

The advanced integral methods are highly recommended and have become popular. For example,

leading up to Jan. 2014, the most popular advanced integral method (AIC) [67] was that proposed

by Vyazovkin in 2001; it was cited more than 400 times on Google Scholar. Most of these

citations relate to obtaining activation energies of various solid reactions, while some use it along

with the Friedman differential (FR) method [56] as a standard to analyze the reliability of other

methods, especially newly developed ones [52, 68, 70, 98, 116, 117]. To the authors’ knowledge

of the literature review, all publications have agreed that the advanced integral methods

(including the differential methods if data do not contain noise) are more accurate than other

isoconversional methods.

The conclusions concerning the accuracy of isoconversional methods correlate directly to the

basic assumption; they may be not reliable if this assumption is problematic. By theoretical and

simulation analyses, this study provides critical information concerning the generally accepted

conclusions about these models and provides some recommendations about using isoconversional

methods to evaluate the activation energies of parallel independent reactions and competitive

reactions.

For brevity, this study focuses on the Friedman (FR) [56], Ozawa-Flynn-Wall (OFW) [72, 73]

and Kissinger-Akahira-Sunose (KAS) [61, 83], and AIC methods because they are

representative and the most popularly-used differential, regular integral, and advanced integral

isoconversional methods, respectively.

5.2 Error Sources of Isoconversional Methods

The overall kinetic equation of a competitive reaction is given as,

65

( ) ( )αβ

αβ

ααα2

221

1121 expexp fRTEAf

RTEA

dTd

dTd

dTd

−+

−=+= (5.1)

Then

221122

11

//

// EfEfE

dTddTdE

dTddTdE +=+=

αα

αα

α (5.2)

where the contribution percent 121 =+ ff .

Some authors [117] suggest the contribution ratio 21 : ff should be dependent on the heating rate,

in which case E is not only a function of α but also of β , i.e.,

( )βα ,EEapp = (5.3)

where appE is defined as the overall or apparent energy for a complex reaction at given α and β.

The same dependence is obtained for other types of complex reactions like parallel independent

reactions. However, the starting point inherent to isoconversional methods is to obtain E as a

function only of α . This is a problematic assumption that would introduce error into any of the

isoconversional methods, and is labeled as “error source 1” (ES1) in the following discussion.

The reason that the AIC and FR methods are considered to provide “exact” values is that they

may use small ranges of α∆ or α∆ close to zero, and so they only suffer from ES1.

Comparatively, the OFW and KAS methods have two additional error sources (ES2 and ES3, as

discussed in the following):

ES2: employing the temperature integral approximation leads to inaccurate values of E; many

analyses [29, 108] have been carried out on this topic. It was concluded that the KAS method

offers significant improvements in accuracy of E values relative to the OFW method because it

uses a more accurate approximation[54];

66

ES3: the integration from 0 to α in Eq. (2.2) assumes E is a constant; otherwise it would

introduce significant systematic errors.

Therefore, the following equation can represent the essence of the differences in error for the AIC

and FR methods relative to the OFW and KAS methods:

3211 ESESESES ++< (5.4)

Figure 5.1 Simulation results obtained from test T1, (beta1-3 are true values for

different βi: it also applies all Figures)

However, the term ES2 + ES3 does not have to increase, a priori, the overall value of

321 ESESES ++ . Actually, in some cases, the source of error for the AIC and FR methods

may be singular and lead to more serious errors than for the OFW and KAS methods.

In general, the isoconversional methods try to obtain a one dimensional parameter αE , usually

labeled as the “apparent”, “overall”, “empirical” or “global” activation energy, to characterize a

80

100

120

140

160

180

0 0.2 0.4 0.6 0.8 1

E k

J/m

ol

α

FRAICOFWKASbeta1beta2beta3

67

two dimensional parameter ( )βα ,E . It is proposed that no isoconversional method can provide

“exact” αE values; instead, the accuracy of an isoconversional method should be judged by its

overall performance when testing one kind of reaction as compared to the overall performance of

other isoconversional methods on the same reaction.

Figure 5.2 Simulation results obtained from test T2

5.3 Simulations and Analysis

5.3.1 Parallel Independent Reaction

A parallel independent reaction is simulated that has two different reactions, as given below:

( ) ( ) ( )2222

111 1exp11exp α

βα

βα

−

−−+−

−=

RTEAc

RTEAc

dTd (5.5)

60

100

140

180

220

260

0 0.2 0.4 0.6 0.8 1

E k

J/m

ol

α


68

where 31−β = 5, 10, 20 K/min, 1A = 15.20 min10 − , 1E =240 kJ/mol, 2A = 18 min10 − , 2E = 100

kJ/mol, and c = 0.1, 0.5, 0.9, for tests T1-T3 respectively. For test T4, c = 0.5, and the value of

1A is 5.1910 . The reactions T1-T3 involve different contributions and are totally or heavily

overlapped. Two tests with partly overlapped reactions are labeled as T5 and T6, for which c =

0.5, and the value of 1A is 5.1810 and 123 min10 − , respectively.


The true values of αE for different iβ that are calculated using Eq. (5.2) and the values

determined by various methods are shown in Figs. 5.1-5.6. The major results from the T1-T6 tests

are presented below:

1. The apparent activation energies have strong dependency on the heating rates. If no dominant

reaction exists - for example, for T2 where c = 0.5, the average differences between 5 K/min and

20 K/min heating rates, calculated by ∑=−=∆

19

1 ,1,3191

i iiEEE ββ , was 14.74 while the maximum

60

100

140

180

220

260

0 0.2 0.4 0.6 0.8 1

E k

J/m

ol

α


69

difference attained was 28.09 kJ/mol. If a dominant reaction existed, the dependence was

relatively weak; for example, for T1 where c = 0.1, the average difference was 8.62 kJ/mol.

2. The use of the FR and AIC methods led to almost the same dependencies of αE on α , as

indicated in the figure by their lines having been merged; the use of the OFW and KAS methods

led to same trends of αE variation with α and was not able to detect if advantages existed for

using the KAS method relative to the OFW method; the accuracy of the temperature integral is

trivial and does not require attention as compared to the errors introduced by the basic assumption

of the isoconversional methods;


3. In all tests except for T4, the behaviors of the OFW and KAS results were better than of the FR

and AIC results in evaluating the dependencies of αE on α for parallel independent reactions.

Specifically, if the reactions were partly overlapped, the use of the OFW and KAS methods was

80

120

160

200

240

280

0 0.2 0.4 0.6 0.8 1

E k

J/m

ol

α


70

better than the use of the FR and AIC methods. If the reactions were totally or heavily

overlapped, the dependences of αE on α obtained by the FR and AIC methods may seriously

deviate from the true values, as is observed in Figs. 5.1-5.3 where dependences obtained by the

FR and AIC methods were far from any of the three true αE lines; in contrast the dependencies

obtained by OFW and KAS methods did not have suffer from this inaccuracy. For T4, the

behavior of the FR and AIC results was a little better than of the OFW and KAS methods, but

they also captured true overall trends of the dependencies of αE on α .


4. The use of the FR and AIC methods led to false values that were smaller or bigger than both

1E and 2E , and even more questionable values were found in simulations using other parameters

80

120

160

200

240

280

0 0.2 0.4 0.6 0.8 1

E k

J/m

ol

α

FR

AIC

OFW

KAS

71

(these are not shown for brevity). In contrast, the use of the OFW and KAS methods never

suffered from these significant problems.

5. When the reactions were partially separated- as for T5 and T6- the αE values were considered

reliable at the beginning and/or ending points for all methods; when the contribution of one of the

reactions was dominant- as for T1 and T3 – the αE values for the dominating reaction could be

obtained by all the methods.

Hence, in summary, the OFW and KAS methods gave better results than the values obtained by

the FR and AIC methods; Eq. (6.4) was not suitable for use with parallel independent reactions;

and, the introduction of ES2 and ES3 for the OFW and KAS methods a flattened or “averaged”

influence decrease the error caused by ES1.


60

100

140

180

220

260

0 0.2 0.4 0.6 0.8 1

E k

J/m

ol

α

FRAICOFWKASture1true2true3

72


5.3.2 Parallel Competitive Reaction

A parallel competitive reaction is simulated that has two different reactions, as given by:

( ) ( )2222

111

21

1exp1exp αβ

αβ

ααα−

−+−

−=

+

=

RTEA

RTEA

dTd

dTd

dTd

(5.6)

where 31−β = 5, 10, 20 K/min, 1A = 15.20 min10 − , 1E = 240 kJ/mol, 2A = 18 min10 − and 2E = 100

kJ/mol. The two reactions in this Test (T7) were equally weighted, which means that in the

overall reaction ( ) ( )21 /~/ dTddTd αα and no dominating reaction existed. Another two tests

were conducted where one dominating reaction existed (T8) that had 1A = 15.19 min10 − ; using this

1A value led to ( ) ( )21 // dTddTd αα << , i.e. the reaction was dominated by reaction 2. The

80

110

140

170

200

0 0.2 0.4 0.6 0.8 1

E k

J/m

ol

α


73

other test with a dominating reaction (T9) had 1A = 122 min10 − ; using this 1A value led to

( ) ( )21 // dTddTd αα >> , i.e. the reaction was dominated by reaction 1.

1. The values of αE had strong dependencies on the heating rates independent of whether a

dominating reaction existed or not: the average αE differences when using 5 K/min, 10 K/min

and 20 K/min for T7-T9 were 30.70, 13.09, and 17.10 kJ/mol, respectively, and the maximum

differences were 34.46, 18.14, 32.13 kJ/mol at α = 0.35, 0.70, 0.05, respectively;

The true values of αE and the values obtained by various methods are shown in Figs. 5.7-5.9. A

discussion of the results from the T7-T9 tests is presented below.


2. The behaviors of the FR and AIC results for evaluating the dependencies of αE on α for

parallel competitive reactions were better than of the OFW and KAS results. The use of the FR

90

100

110

120

130

140

0 0.5 1

E k

J/m

ol

α


74

and AIC methods also led to the same dependencies of αE on α . Comparatively, the use of the

OFW and KAS methods also captured the overall trend of the dependencies but caused over

smoothing of the true values.

In these tests, the dependency of αE on α obtained by the FR and AIC methods were almost

identical to the dependency on true values of 2β . It is worth noting, however, that this observation

does not mean to imply that the same trends could be expected for cases that use different values

and model functions. However, the dependence of αE on α for competitive reactions obtained

by the FR and AIC methods tended to be better than that obtained by the OFW and KAS

methods; this observation then suggests Eq. (6.4) holds for competitive reactions.


140

160

180

200

220

240

0 0.2 0.4 0.6 0.8 1

E k

J/m

ol

α


75

Galway [118] collected 404 sets of E and A values from a wide range of solid state reactions for

which the range of pre-frequency factors and activation energies for a majority of the reactions

were within 30060 ≤≤ E kJ/mol and 196 1010 ≤≤ A 1min− , respectively. During this study,

many simulation tests using various combinations of iβ and ( )αf , and of E and A, were

performed besides those values that were discussed in the previous sections. The overriding result

of these simulations was that the behaviors discussed for reactions T1-T9 also held for these other

simulations. Therefore, it is reasonable to conclude that the results of T1-T9 have common and

important significance.

5.4 Conclusion

Studies of the accuracy of using isoconversional methods to evaluate the activation energy from

non-isothermal solid state reaction data have concluded that the advanced integral

isoconversional methods (and the differential ones if no noise involved) are more accurate than

the regular integral ones. This research suggests this conclusion is problematic; the use of regular

isoconversional methods involve a temperature integral approximation and the assumption that E

is constant over the integration from 0 to α , but has failed to consider the influence of the basic

assumption on isoconversional methods.

This study showed that there are three kinds of error sources involved in various isoconversional

methods: the basic assumption (ES1); the temperature integral approximation (ES2); and the

assumption that E is constant over the integration from 0 to α (ES3). The differential methods,

such as the FR method, and the advanced integral ones, such as the AIC method, suffer from

source ES1; the regular integral methods such as the OFW and KAS methods suffer from all three

(ES1, ES2 and ES3). However, this study for the first time gave critical insight into information

about the generally accepted idea that advanced integral isoconversional methods (and

76

differential ones if the data do not contain noise) are more accurate than the regular integral

isoconversional methods.

Simulations have been carried out for parallel independent reactions and competitive reactions.

The results of these simulations were: the idea that the FR and AIC methods would give exact

values of Eα was not correct; for parallel independent reactions the use of the OFW and KAS

methods was better than the use of FR and AIC methods; for competitive reactions, the use of the

FR and AIC methods was better than the use of the OFW and KAS methods; the error introduced

by the source of temperature integral approximation was much less important than the other two

sources and can be neglected for complex reactions.

Therefore, if one wants to apply an isoconversional method, the following recommendations are

made. Additional studies testing the accuracy of the isoconversional methods or the reliability of

newly developed methods should use the true values of the heating programs rather than the

values determined by the advanced integral methods or the differential ones. Studies should also

give more attention to the basic assumption inherent to the isoconversional methods and the

assumption that E is a constant over the integration range from 0 to α rather than the simplified

the temperature integral. Finally, the use of the AIC and FR methods should yield to the use of

regular integral methods such as the OFW and KAS methods for parallel independent reactions.

Based on theoretical and simulation analyses, it is proposed that the use of advanced methods is

better than regular ones for competitive reactions, but for parallel independent reactions this

research shows for the first time that the use of regular integral methods is better than the use of

the advanced integral ones.

77

CHAPTER 6: CONCLUSION

This study talked in detail of model-fitting, model free and invariant kinetic parameters methods

that have been developed, and proposed a comprehensive method based on previous methods for

obtaining kinetic parameters for solid state reactions. The conclusions are as follows:

Firstly, this study examined the advantages and disadvantages of previously existing methods.

Model-fitting methods are easy to apply and are able to obtain single pair values for the activation

energy and pre-exponential factor. However, model-fitting methods are highly unreliable because

they require an existing knowledge of reaction mechanism, which is difficult to tell in most

situations. Model free methods originated from the isoconversional principle, which is often

called the basic assumption. This assumption provides model free methods - the freedom to avoid

the usage of pre-knowledge of reaction mechanism. Additionally, model free methods are able to

provide variable values of activation energy as a function of reaction degree. Previous studies

comparing the reliability of those methods have not paid attention to the influence of the basic

assumption on model free methods, and therefore, earlier conclusions are problematic. The

invariant kinetic parameters method is also able to provide more reliable single pair values of

activation energy and pre-exponential factor than model-fitting methods, but it is not applicable

for complex reactions where the kinetic parameters such as activation energy vary with reaction

degree and heating programs.

Secondly, this study has determined that the benefits of the invariant kinetic parameters method

are underestimated and an incremental version of the method has been developed. This

incremental method is able to provide values for both the activation energy and pre-exponential

factor for complex reactions. Although those values often deviates heavily from true values, this

study showed that the incremental invariant kinetic parameters method can be used to

successfully determine the reliability of values obtained by model free methods.

Thirdly, based on model free methods and the invariant kinetic parameters method, this study

proposed a comprehensive method. This method was tested on both simulation and experimental

data of the decomposition of calcium carbonate and provided results showing noticeable

advantages over other methods. The comprehensive method can evaluate the reliability of the

results calculated by model free methods, determine the dependence of activation energy on the

heating programs used, select reliable activation energy and provide variable pre-exponential

78

factor values for complex reactions, and to a certain degree, help to judge the quality of the

experimental data.

In addition, this work compares the accuracy of existing model free methods by considering the

influence of the basic assumption. This discussion is helpful for researchers to select a better

isoconversional method that matches with the incremental invariant kinetic parameters method

for different types of complex reactions.

79

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90

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91

VITA YUNQING HAN Previously attended University of Science and Technology of China (Hefei, China), 2008-2011. Publications 1. Han Y., T. Li, and K. Saito, Comprehensive method based on model free method and IKP

method for evaluating kinetic parameters of solid state reactions. Journal of

computational chemistry 2012. 33: p. 2516-2525.

2. Han, Y., T. Li, and K. Saito, A modified Ortega method to evaluate the activation energies

of solid state reactions. Journal of Thermal Analysis and Calorimetry, 2013: p. 1-5.

3. Han, Y., H. Chen, and N. Liu, New incremental isoconversional method for kinetic analysis

of solid thermal decomposition. Journal of thermal analysis and calorimetry, 2011.

104(2): p. 679-683.

theoretical study of thermal analysis kinetics

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